Pith. sign in

REVIEW 3 major objections 4 minor 38 references

Hierarchical Quantum Error Correction with Hypergraph Product Code and Rotated Surface Code

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims a concatenated HGP-surface code can outperform the rotated surface code for size parameter s≥4 and distance d≥25 at physical error rates around or below 10^-2.

desk verdict A practical hierarchical qLDPC-surface scheme with a clever lookup-table soft decoder, but the headline s>=4, d>=25 crossover rests on an extrapolated power-law comparison that needs a direct Monte Carlo check. read the letter →

arxiv 2505.18592 v2 pith:BTREJIZJ submitted 2025-05-24 quant-ph

Junichi Haruna , Keisuke Fujii This is my paper · ORCID
classification quant-ph PACS 03.67.Pp
keywords quantumerrorcorrectionhypergraphproductcoderotatedsurfaceconcatenationbeliefpropagationorderedstatisticsdecodingLDPCcodesfault-tolerantcomputationnearest-neighborarchitecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes a hierarchical quantum error correction code that stacks a random hypergraph product (HGP) code on top of a distance-5 rotated surface code, so the non-local HGP code can be implemented with nearest-neighbor hardware via lattice surgery. The authors claim that with soft-decision decoding using belief propagation with ordered statistics plus syndrome-conditioned logical error probabilities from a lookup table, the concatenated code suppresses logical errors below threshold and removes an error floor. Under a code capacity depolarizing noise model, they derive explicit conditions for the hierarchical code to beat the rotated surface code on both qubit efficiency and logical error rate. The headline result is that at physical error rates around or below $10^{-2}$, the concatenated code wins for size parameter s≥4 (16 logical qubits) and surface distance d≥25, with parameters [[$10^{4}$,16,2α]] where α≈14.38. If right, this gives a resource-efficient, planar-compatible route to fault-tolerant quantum computing.

What carries the argument

The load-bearing object is the concatenated pair: a distance-5 rotated surface code as the lower layer and a (3,4)-random HGP code as the upper layer, whose combined parameters are [[$625s^{2}$, $s^{2}$, 5d_s]] with d_s the largest distance found among 1000 sampled HGP instances. What makes the decoder work is a precomputed lookup table for the L=5 surface code: for every one of the ~3.4×$10^{7}$ X/Z syndromes it stores the most probable error, the set of errors that would cause a logical failure, and the syndrome-conditioned logical error probability P_L(p|s), which is fed as soft information into a belief propagation-plus-ordered-statistics (BP-OS) decoder of depth λ=10 for the upper HGP layer. This soft-decision chain yields the measured scaling law p_L=(p/̄p_th)^{b s^c}, which is the formula that carries the whole crossover argument against the surface code.

What would settle it

Run direct Monte Carlo decoding at p=$10^{-2}$ and p=$10^{-3}$ for the s=4, d=25 concatenated code and compare the measured logical error rate per logical qubit with that of a distance-25 rotated surface code under the same depolarizing noise model; if the concatenated code's error rate is not lower, or if the power-law exponent flattens at s=4, the central claim fails.

Watch

Extended reading notes

Core claim

The central claim is that concatenating a (3,4)-random HGP code with a rotated surface code of distance 5 yields a code that can beat the planar surface code once the lower-layer equivalent has distance d≥25 and the HGP size parameter s≥4. Qubit efficiency requires d≥25 because the concatenated code uses 625 physical qubits per logical qubit, while a d-by-d surface code uses $d^{2}$ per logical qubit. For logical error rate, the paper fits p_L = (p/̄p_th)^{b s^c} with b=5.481 and c=0.667, compares it with the surface-code scaling (p/̄p_s_th)^{(d+1)/2}, and solves for the crossover. The resulting threshold size parameter s_rs(p,d) is nearly flat below pseudo-threshold because the two pseudo-thresholds are close, giving s≥3.65 and hence s≥4. At s=4 the code has parameters [[$625s^{2}$, $s^{2}$, 5d_s]] = [[$10^{4}$,16,2α]] with α≈14.38, versus a single distance-25 surface block [[625,1,25]]; the concatenated code is claimed to match or exceed it in logical error suppression while carrying 16 logical qubits in the same per-logical-qubit footprint.

Load-bearing premise

The conclusion hinges on the assumption that the fitted scaling formula for logical error rates, measured for small sizes, keeps holding unchanged for larger sizes; the paper does not test that point directly at the claimed crossover.

Editorial extensions

If this is right

  • At p around or below 10^-2, a single [[10^4,16,2α]] block with s=4 replaces 16 distance-25 surface-code patches, giving the same 625 physical qubits per logical qubit while matching or improving the logical error rate.
  • For d>25, the hierarchical code becomes strictly more qubit-efficient than the rotated surface code, and the advantage grows with d.
  • Increasing s within one block suppresses logical errors more than preparing s^2 independent copies of the same code, because distance grows with s.
  • The general condition is s ≥ ((d+1)/(2b))^{1/c} and d ≥ 25, so any upper-layer qLDPC code with a higher encoding rate or better distance scaling lowers the required d and s.
  • Changing the lower-layer surface distance L1 to 3 or larger shifts the required HGP distance d_s to 8 or 10 in the hard-decision analysis, so the architecture is tunable.

Reading between the lines

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

  • If the claimed crossover holds, 10^4 physical qubits for 16 logical qubits puts a planar-compatible quantum memory in a range relevant to near-term demonstrations, though the paper's code capacity estimate omits measurement noise and leakage that would raise this overhead.
  • The lookup-table soft-information construction suggests a template for other topological lower layers: any code whose syndrome-conditioned logical error probability can be computed offline can feed a qLDPC upper layer, potentially improving threshold estimates.
  • A direct Monte Carlo comparison at s=4 and d=25, rather than extrapolated scaling curves, would be the natural next check; the authors do not report one, so the concrete crossover point is an extrapolation.
  • Future work could replace the HGP upper layer with asymptotically good qLDPC codes; the paper notes this would lower the distance at which concatenation beats the surface code, implying an entire family of better hierarchical codes.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript proposes a hierarchical quantum error correction scheme in which a (3,4)-random hypergraph product (HGP) code is concatenated with a distance-5 rotated surface code, so that the non-local HGP code can be implemented with nearest-neighbor interactions via lattice surgery. The lower layer is decoded with a lookup table that supplies syndrome-conditioned logical error probabilities, and the upper layer is decoded with belief propagation plus ordered statistics (BP-OS). Under a code-capacity depolarizing noise model, Monte Carlo simulations are used to extract a power-law scaling of the logical error rate, p_L = (p/p_th)^alpha with alpha = b s^c, and the manuscript derives the condition s >= 4 and d >= 25 for the concatenated code to outperform a rotated surface code of distance d in both qubit efficiency and logical error rate. The central claimed operating point is s = 4, giving parameters [[10^4, 16, 2 alpha]] with alpha approximately 14.38, compared with a distance-25 rotated surface code.

Significance. The problem addressed is important: realizing constant-rate qLDPC codes on planar, nearest-neighbor hardware is a key bottleneck for low-overhead fault-tolerant quantum computing, and a concrete numerical study of a concatenated HGP-surface architecture with a soft-decision decoder is a useful contribution. The paper has clear strengths: a substantial Monte Carlo campaign with up to 10^9 samples, an explicit lookup-table decoder with precomputed syndrome-conditioned probabilities, a public data/code repository (DOI 10.5281/zenodo.15660987), and a transparent fitting procedure for the scaling parameters. If the central crossover claim were validated by direct simulation, the result would be practically relevant as a possible resource-efficient alternative to large surface codes. However, as discussed below, the headline condition s >= 4 and d >= 25 rests on an extrapolation of fitted power laws rather than on a direct end-to-end Monte Carlo comparison at the claimed operating point, and the printed probability formulas in Eqs. (20)-(21) contain a normalization error that affects the soft information used by the decoder unless silently corrected in the implementation.

major comments (3)
  1. [III B, Eqs. (20)-(21)] Equations (20) and (21) define Pe(p,s) and Pc(p,s) with factors p^{wt(e)}(1-p)^{wt(e)}. For the L = 5 rotated surface code there are n = 25 data qubits per error type, so the probability of a configuration of weight w is p^w(1-p)^{n-w}. As printed, the two sums are not correctly normalized and systematically mis-weight configurations; if the implemented lookup table follows the text, every syndrome-conditioned logical error probability PL(p|s) fed into the BP decoder is biased. Please correct the second exponent to n - wt(e) and confirm that the posted code uses the corrected expression.
  2. [III C, Eqs. (39)-(43) and Figure 10] The central condition s >= 4 and d >= 25 is derived by substituting the fitted parameters b = 5.481, c = 0.667, and pbar_c^th = 0.157 into Eq. (43) and comparing the resulting power laws with Eq. (40). However, Eqs. (39)-(40) are scale-free power laws without constant prefactors, and Figure 10 plots these fitted formulas rather than Monte Carlo data at the claimed operating point. The crossover is therefore an extrapolation of the same fits used to define the scaling, and it is sensitive to the fit parameters: for example, with c = 0.6 the value of srs(0.01, 25) exceeds 4, which would invalidate the integer condition. The authors' own caveat in Section III B that BP-OS may return incorrect corrections for high-weight errors is relevant precisely in the extrapolated regime. A direct end-to-end Monte Carlo comparison at s = 4, d = 25, or an equivalent validation with uncertainty quantification on all fitted parameters, is needed before the central claim can be accepted.
  3. [III C, Eq. (50)] Equation (50) gives [[10^4, 16, 2 alpha]] with alpha approximately 14.38 for s = 4, but Equation (8) gives [[625 s^2, s^2, 5 d_s]], and Equation (11) gives d_s approximately 2.76 * 4^0.660, which is about 7.0, implying a concatenated distance of roughly 35, not 2 alpha approximately 28.8. Moreover, alpha is already used for the scaling exponent in Eqs. (22)-(23), so the reuse of alpha in Eq. (50) is confusing. Please state the actual measured distance of the selected HGP instance for s = 4 and reconcile Eq. (8), Eq. (11), and Eq. (50).
minor comments (4)
  1. [III C / Figure 10] The sentence before Eq. (50) says 'In Figure 7, we plot logical error rate...', but Figure 7 compares error scaling factors; the logical-error-rate comparison appears in Figure 10. Please correct the cross-reference.
  2. [III C, Eqs. (43)-(48)] The function introduced as srs(p,d) in Eq. (43) is later written as s(p,d) in Eqs. (45)-(48); please unify the notation.
  3. [III B] The phrase 'rigorously compute syndrome-conditioned logical error probabilities rigorously' contains a duplicated adverb; please remove one occurrence.
  4. [III A / Figure 3] The caption of Figure 3 refers to a 'blue line' and an 'orange line', while the surrounding text describes data points and a fitting line; please check that the colors and line styles are described consistently with the figure.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the s≥4, d≥25 crossover follows from independently fitted Monte Carlo scaling laws and an external surface-code threshold, not from the conclusion itself.

full rationale

The central claim (Eq. 49) is not circular. The paper obtains b and c in Eq. (23) by fitting the scaling exponents extracted from independent Monte Carlo simulations of the concatenated code at s = 2 through 6, and it obtains the rotated surface-code pseudo-threshold from an external source, Ref. [33]. Equations (39)-(43) are then an analytic comparison of those two independently determined scaling laws; the condition s ≥ 4 and d ≥ 25 is the solution of the inequality, not an input to the fits. The comparison uses the fitted power law at s = 4, which is inside the simulated range, so the claim is a comparison of measured/fitted behavior rather than an extrapolation of fitted parameters to an unseen regime. No load-bearing self-citation or imported uniqueness theorem appears. The only notable concern is a likely typographical normalization error in Eqs. (20)-(21), where the configuration probability is written as p^{wt(e)}(1-p)^{wt(e)} instead of p^{wt(e)}(1-p)^{n-wt(e)}; if the implemented lookup table follows the text, the syndrome-conditioned probabilities would be incorrectly normalized. That is a correctness issue affecting the numerical results, not a circularity of the derivation chain, and it does not make the outperformance condition definitionally equivalent to its inputs.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The central claim rests on fitted power laws, a code capacity noise model, and the representative behavior of randomly sampled classical codes. No new physical entities are introduced.

free parameters (4)
  • distance fit exponent pair (b_h, c_h) = b_h=2.76, c_h=0.660
    Fit to maximum observed code distance d_s = b_h*s^c_h from 1000 samples per s, Eq. (11). Used for hard-decision scaling and for estimating d_s for larger s.
  • logical error scaling exponent alpha = b*s^c = b=5.481, c=0.667
    Least-squares power-law fit to Monte Carlo logical error rates, Eq. (23). Directly enters Eq. (43) through Eq. (48), so the s>=4 condition depends on these fitted values.
  • average pseudo-threshold pbar_c^th = 0.157
    Average of fitted pseudo-thresholds over size parameters, Fig. 6. Used as the normalization in the logical error scaling formula, Eq. (39).
  • surface-code pseudo-threshold pbar_s^th = 0.1776
    Imported from Ref. [33], not fitted in this paper. Used as the comparison threshold for the rotated surface code in Eq. (40).
assumptions (3)
  • ad hoc to paper The logical error rate follows p_L=(p/pth)^alpha below the pseudo-threshold.
    Assumed in Eq. (22) and used for all conditions. It is an empirical fit, not a theorem, and no error bars or direct validation at the crossover point are provided.
  • domain assumption Code capacity depolarizing noise with independent, identically distributed errors and no measurement errors.
    Eq. (14) and Section III.B set this model; all numerical claims live inside it, so circuit-level noise and measurement errors are outside the demonstrated result.
  • domain assumption Random (3,4) classical parity check matrices generically have k_c=s logical bits and k_T_c=0, and the maximum observed distance d_s is representative.
    Section III.A, Eq. (12), and Fig. 3 report this from sampled instances. The construction selects the best of 1000 instances, and later hard-decision estimates also assume d_s is even for all s.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Hierarchical Quantum Error Correction with Hypergraph Product Code and Rotated Surface Code." pith.science (2026). https://pith.science/paper/BTREJIZJ

@misc{pith2026250518592,
  author       = {Pith},
  title        = {Pith review of: Hierarchical Quantum Error Correction with Hypergraph Product Code and Rotated Surface Code},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BTREJIZJ}},
  note         = {Machine review of arXiv:2505.18592}
}
abstract

We propose and analyze a hierarchical quantum error correction (QEC) scheme that concatenates hypergraph product (HGP) codes with rotated surface codes, which is compatible with quantum computers with only nearest-neighbor interactions. The upper layer employs (3,4)-random HGP codes, known for their constant encoding rate and favorable distance scaling, while the lower layer consists of a rotated surface code with distance 5, allowing hardware compatibility through lattice surgery. To address the decoding bottleneck, we utilize a soft-decision decoding strategy that combines belief propagation with ordered statistics (BP-OS) decoding, enhanced by a syndrome-conditioned logical error probability computed via a tailored lookup table for the lower layer. Numerical simulations under a code capacity noise model demonstrate that our hierarchical codes achieve logical error suppression below the threshold. Furthermore, we derive explicit conditions under which the proposed codes surpass surface codes in both qubit efficiency and error rate. In particular, for the size parameter $s \geq 4$ (which corresponds to 16 logical qubits) and the distance $d\geq 25$, our construction outperforms the rotated surface code in practical regimes with physical error rates around or less than $10^{-2}$. These results suggest that concatenated qLDPC-surface architectures offer a scalable and resource-efficient path toward near-term fault-tolerant quantum computation.

Figures

Figures reproduced from arXiv: 2505.18592 by the authors.

Figure 1
Figure 1. FIG. 1. Illustration of the rotated surface code of the size [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Illustration of the concatenated code: the lower layer uses a rotated surface code, and the upper layer employs an HGP [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. For later use, we perform a fitting analysis of the code distances against the size parameter. Assuming the following formula ds = bh · s ch , we get bh = 2.76, ch = 0.660. (11) This behavior ( [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Logical error rates of concatenated HGP codes as a [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Pseudo-threshold [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison of error scalings. The blue/orange/green [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: illustrates srs(p, d) as a function of the rotated surface code distance d for different physical error rates p = 10−2 , 10−3 , 10−4 . Summary of Conditions in Soft Decision Case Combining Equation (31) and Equation (43), the con￾catenated code outperforms the rotated …
Figure 9
Figure 9. Figure 9: FIG. 9. Minimum size parameter [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Comparison of logical error rates [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

38 extracted references · 37 canonical work pages

  1. [1]

    Each stabilizer acts on a constant number of phys- ical qubits

    work page

  2. [2]

    Hierarchical Quantum Error Correction with Hypergraph Product Code and Rotated Surface Code

    Each physical qubit participates in a constant num- ber of stabilizers. The high encoding rate of the qLDPC codes is expected to reduce the qubit overhead required for FTQC. In ad- dition, qLDPC codes exhibit favorable scaling proper- ties, such as the code distance increasing with the num- ber of physical qubits. For example, hypergraph product (HGP) cod...

    work page Pith review arXiv 2025

  3. [3]

    Decoding each lower-layer code block using syn- drome extraction and error correction

    work page

  4. [4]

    Although computationally efficient, this method reduces the effective code distance to: deff = 2 d1 + 1 2 d2 + 1 2 , (7) 4 FIG

    Performing syndrome extraction for the upper layer after applying recovery operations at the lower layer, followed by upper-layer decoding based on these syndromes. Although computationally efficient, this method reduces the effective code distance to: deff = 2 d1 + 1 2 d2 + 1 2 , (7) 4 FIG. 1. Illustration of the rotated surface code of the size L = 5. T...

    work page

  5. [5]

    A.Yu. Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1):2–30, January 2003

    work page 2003

  6. [6]

    Fowler, Matteo Mariantoni, John M

    Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. Surface codes: Towards prac- tical large-scale quantum computation. Physical Review A, 86(3), September 2012

    work page 2012

  7. [7]

    Topological quantum memory

    Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. Topological quantum memory. Journal of Math- ematical Physics, 43(9):4452–4505, September 2002

    work page 2002

  8. [8]

    Fowler, Ashley M

    Austin G. Fowler, Ashley M. Stephens, and Peter Groszkowski. High-threshold universal quantum compu- tation on the surface code. Physical Review A, 80(5), November 2009

    work page 2009

Show all 38 references
  1. [9]

    Topologi- cal fault-tolerance in cluster state quantum computation

    R Raussendorf, J Harrington, and K Goyal. Topologi- cal fault-tolerance in cluster state quantum computation. New Journal of Physics, 9(6):199–199, June 2007

    2007

  2. [10]

    Superconducting quantum circuits at the surface code threshold for fault tolerance

    R Barends, J Kelly, A Megrant, A Veitia, D Sank, E Jef- frey, TC White, J Mutus, AG Fowler, B Campbell, et al. Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature, 508(7497):500–503, 2014

    2014

  3. [11]

    Cross, A.D

    Maika Takita, Andrew W. Cross, A.D. C´ orcoles, Jerry M. Chow, and Jay M. Gambetta. Experimental demonstra- tion of fault-tolerant state preparation with supercon- ducting qubits. Physical Review Letters, 119(18), Octo- ber 2017

    2017

  4. [12]

    Quantum error correction below the sur- face code threshold

    Rajeev Acharya, Laleh Aghababaie-Beni, Igor Aleiner, Trond I Andersen, Markus Ansmann, Frank Arute, Ku- nal Arya, Abraham Asfaw, Nikita Astrakhantsev, Juan Atalaya, et al. Quantum error correction below the sur- face code threshold. arXiv preprint arXiv:2408.13687, 2024

    2024 arXiv

  5. [13]

    AbuGhanem

    M. AbuGhanem. Ibm quantum computers: Evolution, performance, and future directions, 2024

    2024

  6. [14]

    Bombin and M

    H. Bombin and M. A. Martin-Delgado. Optimal re- sources for topological two-dimensional stabilizer codes: Comparative study. Physical Review A, 76(1), July 2007

    2007

  7. [15]

    On the iterative decod- ing of sparse quantum codes, 2008

    David Poulin and Yeojin Chung. On the iterative decod- ing of sparse quantum codes, 2008

    2008

  8. [16]

    Quantum ldpc codes with positive rate and minimum distance propor- tional to the square root of the blocklength

    Jean-Pierre Tillich and Gilles Zemor. Quantum ldpc codes with positive rate and minimum distance propor- tional to the square root of the blocklength. IEEE Trans- actions on Information Theory, 60(2):1193–1202, Febru- ary 2014

    2014

  9. [17]

    Good quantum ldpc codes with linear time de- coders, 2022

    Irit Dinur, Min-Hsiu Hsieh, Ting-Chun Lin, and Thomas Vidick. Good quantum ldpc codes with linear time de- coders, 2022

    2022

  10. [18]

    Asymptotically good quantum and locally testable classical ldpc codes, 2022

    Pavel Panteleev and Gleb Kalachev. Asymptotically good quantum and locally testable classical ldpc codes, 2022

    2022

  11. [19]

    A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes

    Sergey Bravyi and Barbara Terhal. A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes. New Journal of Physics , 11(4):043029, April 2009

    2009

  12. [20]

    Evered, Alexandra A

    Dolev Bluvstein, Simon J. Evered, Alexandra A. Geim, Sophie H. Li, Hengyun Zhou, Tom Manovitz, Sepehr Ebadi, Madelyn Cain, Marcin Kalinowski, Dominik Hangleiter, J. Pablo Bonilla Ataides, Nishad Maskara, Iris Cong, Xun Gao, Pedro Sales Rodriguez, Thomas Karolyshyn, Giulia Seme...

    2023

  13. [21]

    Beverland, and Maxime A

    Nicolas Delfosse, Michael E. Beverland, and Maxime A. Tremblay. Bounds on stabilizer measurement circuits and obstructions to local implementations of quantum ldpc codes, 2021

    2021

  14. [22]

    Pattison, Anirudh Krishna, and John Preskill

    Christopher A. Pattison, Anirudh Krishna, and John Preskill. Hierarchical memories: Simulating quantum ldpc codes with local gates, 2023

    2023

  15. [23]

    Pattison, and John Preskill

    Nadine Meister, Christopher A. Pattison, and John Preskill. Efficient soft-output decoders for the surface code, 2024

    2024

  16. [24]

    Concatenate codes, save qubits, 2024

    Satoshi Yoshida, Shiro Tamiya, and Hayata Yamasaki. Concatenate codes, save qubits, 2024

    2024

  17. [25]

    Ldpc-cat codes for low-overhead quantum computing in 2d, 2024

    Diego Ruiz, J´ er´ emie Guillaud, Anthony Leverrier, Maz- yar Mirrahimi, and Christophe Vuillot. Ldpc-cat codes for low-overhead quantum computing in 2d, 2024

    2024

  18. [26]

    Kaufman, and Andrew Lucas

    Yifan Hong, Matteo Marinelli, Adam M. Kaufman, and Andrew Lucas. Long-range-enhanced surface codes. Physical Review A, 110(2), August 2024

    2024

  19. [27]

    Optimal and efficient decoding of concate- nated quantum block codes

    David Poulin. Optimal and efficient decoding of concate- nated quantum block codes. Physical Review A, 74(5), November 2006

    2006

  20. [28]

    White, Simon Burton, and Earl Campbell

    Joschka Roffe, David R. White, Simon Burton, and Earl Campbell. Decoding across the quantum low-density parity-check code landscape. Physical Review Research, 2(4), December 2020

    2020

  21. [29]

    A. R. Calderbank and Peter W. Shor. Good quan- tum error-correcting codes exist. Physical Review A, 54(2):1098–1105, August 1996

    1996

  22. [30]

    Peter W. Shor. Fault-tolerant quantum computation, 1997

    1997

  23. [31]

    Concatenated quantum codes, 1996

    Emanuel Knill and Raymond Laflamme. Concatenated quantum codes, 1996

    1996

  24. [32]

    Fault-tolerant quantum computation with constant error rate, 1999

    Dorit Aharonov and Michael Ben-Or. Fault-tolerant quantum computation with constant error rate, 1999

    1999

  25. [33]

    Emanuel Knill, Raymond Laflamme, and Wojciech H. Zurek. Resilient quantum computation: error models and thresholds. Proceedings of the Royal Society of Lon- don. Series A: Mathematical, Physical and Engineering Sciences, 454(1969):365–384, January 1998

    1969

  26. [34]

    Surface code quantum computing by lattice surgery

    Dominic Horsman, Austin G Fowler, Simon Devitt, and Rodney Van Meter. Surface code quantum computing by lattice surgery. New Journal of Physics, 14(12):123011, December 2012

    2012

  27. [35]

    A game of surface codes: Large-scale quantum computing with lattice surgery

    Daniel Litinski. A game of surface codes: Large-scale quantum computing with lattice surgery. Quantum, 3:128, March 2019

    2019

  28. [36]

    Gallager

    Robert G. Gallager. Low-Density Parity-Check Codes. The MIT Press, September 1963

    1963

  29. [37]

    Multi-path summation for decoding 2d topological codes

    Ben Criger and Imran Ashraf. Multi-path summation for decoding 2d topological codes. Quantum, 2:102, October 2018

    2018

  30. [38]

    Data and code for ”hierarchical quantum error correction with hypergraph product code and rotated surface code”, June 2025

    Junichi Haruna and Keisuke Fujii. Data and code for ”hierarchical quantum error correction with hypergraph product code and rotated surface code”, June 2025

    2025

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.