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REVIEW 4 major objections 5 minor 58 references

Bias-tailored single-shot quantum LDPC codes

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper builds a hierarchy of bias-tailored single-shot quantum codes whose trimmed variants use 1/6 fewer physical qubits, half the stabilizer measurements, and a quadratically larger minimum distance.

desk verdict Interesting construction, but the advertised single-shot guarantees rest on a wrong lemma proof and a quantifier error in the RSH soundness argument; worth refereeing with major revision expected. read the letter →

arxiv 2507.02239 v1 pith:KJVWZ4DR submitted 2025-07-03 quant-ph

Shixin Wu , Todd A. Brun , Daniel A. Lidar This is my paper
classification quant-ph MSC 81P7094B05 PACS 03.67.Pp
keywords bias-tailoredquantumcodessingle-shoterrorcorrectionLDPChypergraphproductsyndromeencodingXZZXsurfacecodesoundnesschaincomplexes
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 combines two ideas in quantum error correction: tailoring a code to hardware whose bit-flip ($X$) and phase-flip ($Z$) error rates differ, and correcting both data and measurement errors in a single readout round. The authors construct a bias-tailored syndrome-encoded hypergraph product (BSH) code that they claim keeps a single-shot guarantee for every noise model while improving thresholds when $X$ and $Z$ errors are asymmetric. They then trim blocks of stabilizers to obtain simplified and reduced variants: the simplified code uses $5n^4$ physical qubits instead of $6n^4$, halves the stabilizer measurements from $8n^4$ to $4n^4$, and raises the minimum distance from $d$ to $d^2$; the reduced code achieves the same hardware savings but keeps single-shot protection only under balanced (depolarizing) noise. A three-dimensional XZZX surface code is shown to be an explicit member of the simplified family. If these claims hold, code designers gain a tunable menu that trades hardware overhead against the expected noise bias.

What carries the argument

The load-bearing object is the syndrome-encoded hypergraph product (SEHGP) code, a CSS code built by taking the homological tensor product of two syndrome-encoded chain complexes, each assembled from two classical codes; the code's stabilizer generators and syndrome checks are read off from the boundary maps of the resulting four-step complex. Soundness of these boundary maps --- meaning small measured syndromes are guaranteed to come from small error operators, up to stabilizers --- is what yields single-shot correction. A second ingredient is the commutation-preserving block rotation (CPHR), a block swap between $X$- and $Z$-type stabilizer columns that preserves the symplectic commutation condition and rearranges the code into disjoint copies of classical codes under pure $X$ or $Z$ noise. The simplified and reduced families are obtained by deleting selected blocks of these boundary maps, and the code parameters follow from the direct-product structure of the kernels: codewords are matrices whose rows and columns lie in the constituent classical codes, giving the $d \to d^2$ distance growth.

What would settle it

Test the inheritance lemma directly: for the binary closed-loop repetition code with parity-check matrix $H$, compute the minimal preimage weight for every syndrome $q$ of weight at most $d$ in the map $H \otimes I$. If any such syndrome requires a preimage whose weight exceeds $|q|^2/4$, the lemma is false and the pure-$X$/$Z$ single-shot claim for the simplified family is falsified.

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Extended reading notes

Core claim

Starting from the syndrome-encoded hypergraph product (SEHGP) code --- obtained by tensoring two syndrome-encoded chain complexes built from four classical linear codes --- the authors apply a commutation-preserving block rotation that swaps $X$- and $Z$-type stabilizer blocks, producing the bias-tailored BSH code. They argue that the SEHGP composite syndrome map is $(t,f)$-sound with $t$ equal to the minimum of the base-code distances and $f(x)=x^2/4$, which by the soundness-to-single-shot theorem makes the family single-shot under depolarizing noise; by soundness of the individual maps, the BSH variant keeps this guarantee under pure $X$ or pure $Z$ noise. Deleting selected stabilizer blocks gives the simplified family, which uses $5n^4$ physical qubits instead of $6n^4$, halves the number of checks to $4n^4$, and increases the minimum distance from $d$ to $d^2$; the bias-tailored simplified code preserves the pure-$X$/$Z$ soundness but not a depolarizing guarantee. The reduced family uses the same hardware savings, sacrifices pure-$X$/$Z$ single-shot protection, and retains soundness under depolarizing noise. The paper works out the simplified case explicitly and identifies the resulting code as a three-dimensional XZZX code on a cubic lattice with parameters $[[5n^4, 2, n^2]]$.

Load-bearing premise

The trimmed codes' single-shot soundness claims depend on the assumption that tensoring a sound syndrome map with an identity matrix preserves the $(t,f)$-soundness property; if this inheritance fails, the pure-$X$/$Z$ single-shot guarantees for the simplified and reduced families collapse.

Editorial extensions

If this is right

  • The full BSH code keeps a single-shot guarantee for every noise model and raises the threshold relative to the untailored SEHGP code whenever $X$ and $Z$ error rates are unequal.
  • The simplified and reduced families use $5n^4$ physical qubits and $4n^4$ stabilizer checks, saving one-sixth of the qubits and half the measurements of the SEHGP code.
  • The simplified family's minimum distance is $d^2$ rather than $d$, so the same logical protection can be reached with smaller physical hardware.
  • Under purely $X$ or purely $Z$ noise, the bias-tailored simplified and reduced codes split into independent copies of smaller classical codes, which is the structural reason for the expected threshold gains.
  • The three-dimensional XZZX code is an explicit member of the simplified family with parameters $[[5n^4,2,n^2]]$, providing a concrete lattice realization.

Reading between the lines

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

  • If the hierarchy is correct, the 1/6 qubit saving and the halving of stabilizer measurements are especially valuable in settings where measurement speed and readout fidelity dominate overhead, since single-shot correction avoids repeated measurement rounds.
  • The reduced code's soundness argument relies on a mixed bound that guarantees at least one of two reduced variants inherits soundness; the same mechanism might extend the guarantee to intermediate bias levels, not only the two extremes of pure and depolarizing noise.
  • A careful check of the inheritance lemma used for the pure-$X$/$Z$ soundness claims would settle whether the simplified family keeps its single-shot guarantee under strongly biased noise or only under depolarizing noise.
  • A numerical decoder benchmark on the 3D XZZX example would directly test whether the predicted biased-noise thresholds materialize; the paper leaves simulation for future work.
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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

4 major / 5 minor

Summary. The paper proposes a hierarchy of quantum LDPC codes built from syndrome-encoded hypergraph products and Hadamard rotations. Under the assumption that four base classical codes are identical, it derives parameters for the full SEHGP family ([[6n^4,6k^4,d]], 8n^4 checks), the simplified SSH/BSSH family ([[5n^4,2k^4,d^2]], 4n^4 checks), and the reduced RSH/BRSH family ([[5n^4,4k^4,d]], 4n^4 checks). It claims single-shot soundness for the bias-tailored BSH code under all noise models, pure-X/Z soundness for BSSH, and depolarizing soundness for RSH/BRSH, and it presents a 3D XZZX code as an explicit BSSH example. The arguments are analytic and rely on Campbell's soundness framework for single-shot error correction.

Significance. If the claims were established, the hierarchy would provide a useful set of tunable trade-offs between hardware overhead, code distance, and noise bias, and the 3D XZZX example is a concrete and appealing illustration. The parameter calculations are self-contained and give explicit numerical savings. However, two load-bearing proofs are invalid as written (Lemma III.2 and Appendix D), the single-shot distance for the RSH codes is not supplied, and the advertised threshold improvements are only conjectured in the text. These issues directly affect the abstract's headline guarantees, so the paper in its present form does not support its central claims.

major comments (4)
  1. [Section III.E, Lemma III.2] The proof of Lemma III.2 is incorrect. After decomposing q=Σ_i α_i⊗e_i and choosing i0, soundness of ∂ provides a preimage F_{i0} with ∂F_{i0}=α_{i0}; it does not provide an element of ker∂. The text instead chooses γ_{i0}∈ker∂ and sets r=γ_{i0}⊗e_{i0}, which satisfies (∂⊗I)r=0 rather than (∂⊗I)r=q. The displayed bound is therefore about a kernel element, not a preimage of q. The lemma may be repairable by taking preimages for all components and using convexity of f(x)=x^2/4, but as written the proof does not establish the tensor-product soundness on which the BSSH pure-X/Z soundness claim relies.
  2. [Appendix D] The proof of good soundness for the RSH code contains a quantifier error. After Corollary D.1, the text says 'Assume, without loss of generality, that the bound |r_b|≤|sL1||sR2| holds' and proves soundness for RSH1, with RSH2 treated as symmetric. But Corollary D.1 only shows that for each syndrome at least one of the two mixed bounds holds; it does not show that the bound corresponding to a fixed code holds for all syndromes of that code. An RSH1 decoder sees only (sL1,sR2), while the alternative bound involves (sL2,sR1). The weight pattern (|sL1|,|sR1|,|sL2|,|sR2|)=(1,10,10,1) with |r_b|=5 satisfies both unmixed bounds and only the second mixed bound, so the WLOG step is invalid. Consequently, Appendix D establishes at most a disjunctive soundness property of the pair of codes, not soundness of either RSH1 or RSH2.
  3. [Section III.G, Single-shot Distance] The text states 'we cannot give the single-shot distances for the RSH codes' because no closed form for H_RS1/H_RS2 is available. Theorem III.1 defines p=min{d_s,t}/2, so without a value or lower bound on d_s, even a correct soundness proof would not imply the (p,q,f) single-shot guarantee. The abstract's claim that the reduced code remains single-shot under balanced, or depolarizing, noise is therefore unproven regardless of the Appendix D issue.
  4. [Abstract vs. Sections III.E and III.G] The abstract states as established results that the BSH code boosts the threshold whenever X and Z errors are asymmetric and that the simplified code's biased-noise threshold is unchanged. In the text these are explicitly conjectural: Section III.E says 'we expect the BSH family to inherit a higher threshold... numerical verification is left for future work,' and Section III.G says the bias-tailored versions are 'expected to outperform' and 'should' have higher thresholds. Threshold claims need either proof or numerical support, or the abstract must be revised to present them as expectations.
minor comments (5)
  1. [Definition III.5] Equation (44) writes the right-hand side as min{∂(E):∂(E)=∂(F), F∈F^n}; this should be min{|F|: ∂(F)=∂(E)}, since the quantity being minimized is the weight of a preimage, not a boundary vector.
  2. [Section III.G] The first paragraph says the SEHGP code is a [[6n^4,4k^4,d]] code, which contradicts Section III.A and Section IV, where the logical-qubit count is 6k^4. Since the RSH logical-qubit count is then described as one-third fewer than 6k^4, the 4k^4 entry is a typo that should be corrected.
  3. [Section III.D] The phrase 'T1 CHPR' should read 'T1 CPHR'; the acronym for commutation-preserving Hadamard rotation is introduced as CPHR.
  4. [Section III.G] The sentence 'we cannot give the single-shot distances for the RSH codes' should use 'for H_RS1 and H_RS2' rather than 'to', and the notation H_RSH1/H_RS1 could be defined more explicitly to distinguish the stabilizer matrix from its syndrome-encoding matrix.
  5. [Appendix D, Lemma D.1] Lemma D.1 says its proof is identical to Lemma 7 of [29], but the lemma is central to the RSH soundness claim. It should either be proved in full or accompanied by a precise statement of how the hypotheses and conclusion map onto Lemma 7 of [29].

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the code parameters, soundness claims, and threshold statements are derived from explicit counting, cited soundness lemmas, and clearly labeled expectations, not from fitted inputs or self-referential definitions.

full rationale

The paper's central claims are built by direct algebraic construction. The BSH code is defined by applying two explicit commutation-preserving Hadamard rotations to the SEHGP stabilizer matrix, and its length, logical dimension, and distance are computed by dimension counting and Betti-number/Kunneth arguments (Eqs. 38-41 and 87), not assumed. Soundness is inherited from Campbell's Lemma III.1 with t equal to the minimum of the base-code distances and f(x)=x^2/4; Corollary III.1 derives the SEHGP/BSH soundness from that lemma rather than from a fitted parameter or from the target property itself. The simplified SSH/BSSH claims (5n^4 qubits, 4n^4 checks, distance d^2) follow from the direct-product-code Lemma II.1 and explicit matrix dimensions, and the paper explicitly flags that certain threshold comparisons are expectations rather than derived predictions ('we expect', 'numerical verification is left for future work'), so no fitted-input-called-prediction pattern is present. The reduced RSH/BRSH soundness discussion leans on Appendix D and on Lemma 7 of Campbell [29]; whether that appendix proves soundness for a fixed RSH code is a correctness and quantifier issue, not a circularity issue, and no argument in the paper reduces the claim to its own assumption. The 3D XZZX example is presented as an explicit instance of the SSH construction after substituting the closed-loop repetition code; this is an identification and equivalence, not a renaming of a known result as a new derivation. The only self-citation in the vicinity ([27], with author Brun) appears in the introduction as background on redundant measurements and is not load-bearing for any theorem. No uniqueness theorem from the authors is invoked, and no ansatz is smuggled in via citation: the Hadamard-rotation rule is stated and verified algebraically in the text, and the base code H_rep for the example is stated as a construction input, not as a consequence of the claims. The derivation chain is therefore self-contained, and the circularity score is 0.

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

The proofs rely on standard algebraic topology and prior soundness lemmas; no free parameters are fitted. The main ad hoc assumptions are the identical-base-code simplification used for resource counts and the unproven soundness-inheritance lemma.

assumptions (4)
  • ad hoc to paper All four base codes are identical [n,k,d] codes with kerH = kerH^T and distance linear in n.
    Section III A; used to derive the reported parameter counts (6n^4, 5n^4, d^2, etc.). The abstract presents these as general facts.
  • domain assumption Constituent classical codes satisfy the orthogonality condition H1 H2^T = 0.
    Section III D, required for the HGP symplectic condition and CPHR commutation.
  • ad hoc to paper The soundness-inheritance lemma (Lemma III.2) holds.
    The proof given is invalid; the lemma is stated as a theorem but uses a kernel element instead of a preimage, so the claim is currently unsupported.
  • domain assumption Noise model is an independent Pauli channel with bias parameter.
    Section III, standard in QEC but restricts applicability of the claims.

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Cite this review

Pith. "Pith review of Bias-tailored single-shot quantum LDPC codes." pith.science (2026). https://pith.science/paper/KJVWZ4DR

@misc{pith2026250702239,
  author       = {Pith},
  title        = {Pith review of: Bias-tailored single-shot quantum LDPC codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KJVWZ4DR}},
  note         = {Machine review of arXiv:2507.02239}
}
abstract

Quantum hardware rarely suffers equal amounts of bit-flip ($X$) and phase-flip ($Z$) errors; one type is often much more common than the other. A code that is ``bias-tailored'' can exploit this imbalance, lowering the fault-tolerance overhead. A complementary idea, called "single-shot" error correction, aims to recover from data errors and noisy measurements in a single round of stabilizer readout, avoiding slow repetition cycles. In this work, we combine these two ideas and build a hierarchy of new quantum codes. The full construction starts from the syndrome-encoded hypergraph product code and then tailors it to the dominant error type. The resulting code keeps the single-shot guarantee for every noise model while boosting the threshold whenever $X$ and $Z$ errors are asymmetric. By removing carefully chosen blocks of stabilizers we obtain two trimmed variants. The first, called the simplified code, cuts the physical-qubit count by $1/6$ and halves the number of stabilizer measurements, yet its minimum distance grows quadratically compared to the standard design and its biased noise threshold is unchanged. The second, called the reduced code, achieves the same hardware savings but trades away single-shot protection for purely $X$ or purely $Z$ noise; instead it remains single-shot under balanced, or depolarizing, noise. In settings where strongly biased noise is likely, either trimmed code offers a less resource-intensive alternative to the full construction. As a concrete illustration, we lift the two-dimensional XZZX surface code to a three-dimensional cubic lattice and show that this ``3D XZZX'' code is an explicit member of the simplified family. Taken together, these bias-tailored single-shot codes provide an adjustable set of code design alternatives, allowing tradeoffs between hardware overhead and noise types.

Figures

Figures reproduced from arXiv: 2507.02239 by the authors.

Figure 1
Figure 1. FIG. 1. Flowchart of the construction of the SEHGP code, [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Layout of the three-dimensional XZZX code. Qubits [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

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Reference graph

Works this paper leans on

58 extracted references · 36 canonical work pages

  1. [1]

    Aliferis, F

    P. Aliferis, F. Brito, D. P. DiVincenzo, J. Preskill, M. Steffen, and B. M. Terhal, Fault-tolerant computing with biased-noise superconducting qubits: a case study, New Journal of Physics11, 013061 (2009)

    work page 2009

  2. [2]

    Lescanne, M

    R. Lescanne, M. Villiers, T. Peronnin, A. Sarlette, M. Delbecq, B. Huard, T. Kontos, M. Mirrahimi, and Z. Leghtas, Exponential suppression of bit-flips in a qubit encoded in an oscillator, Nature Physics16, 509 (2020)

    2020

  3. [3]

    Chamberland, K

    C. Chamberland, K. Noh, P. Arrangoiz-Arriola, E. T. Campbell, C. T. Hann, J. Iverson, H. Putterman, T. C. Bohdanowicz, S. T. Flammia, A. Keller,et al., Building a fault-tolerant quantum computer using concatenated cat codes, PRX Quantum3, 010329 (2022)

    2022

  4. [4]

    P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Physical review A52, R2493 (1995)

    1995

  5. [5]

    A. M. Steane, Error correcting codes in quantum theory, Physical Review Letters77, 793 (1996)

    1996

  6. [6]

    A. R. Calderbank, E. M. Rains, P. M. Shor, and N. J. A. Sloane, Quantum error correction via codes overgf(4), IEEE Transactions on Information Theory44, 1369 (1998)

    work page 1998

  7. [7]

    Lidar and T

    D. Lidar and T. Brun, eds.,Quantum Error Correction (Cambridge University Press, Cambridge, UK, 2013)

    2013

  8. [8]

    Aliferis and J

    P. Aliferis and J. Preskill, Fault-tolerant quantum com- putation against biased noise, Physical Review A78, 052331 (2008)

    work page 2008

Show all 58 references
  1. [9]

    S. Puri, L. St-Jean, J. A. Gross, A. Grimm, N. E. Frat- tini, P. S. Iyer, A. Krishna, S. Touzard, L. Jiang, A. Blais, et al., Bias-preserving gates with stabilized cat qubits, Science advances6, eaay5901 (2020)

    2020

  2. [10]

    J. P. Bonilla Ataides, D. K. Tuckett, S. D. Bartlett, S. T. Flammia, and B. J. Brown, The xzzx surface code, Na- ture communications12, 2172 (2021)

    2021

  3. [11]

    Bombin, R

    H. Bombin, R. S. Andrist, M. Ohzeki, H. G. Katzgraber, and M. A. Martin-Delgado, Strong resilience of topologi- cal codes to depolarization, Physical Review X2, 021004 (2012)

    2012

  4. [12]

    Roffe, L

    J. Roffe, L. Z. Cohen, A. O. Quintavalle, D. Chandra, and E. T. Campbell, Bias-tailored quantum ldpc codes, Quantum7, 1005 (2023)

    2023

  5. [13]

    A. Dua, A. Kubica, L. Jiang, S. T. Flammia, and M. J. Gullans, Clifford-deformed surface codes, arXiv preprint arXiv:2201.07802 (2022)

    2022 arXiv

  6. [14]

    Q. Xu, N. Mannucci, A. Seif, A. Kubica, S. T. Flam- mia, and L. Jiang, Tailored xzzx codes for biased noise, Physical Review Research5, 013035 (2023)

    2023

  7. [15]

    Huang, A

    E. Huang, A. Pesah, C. Chubb, M. Vasmer, and A. Dua, Tailoring three-dimensional surface codes for biased noise, inAPS March Meeting Abstracts, Vol. 2022 (2022) pp. T40–012

    2022

  8. [16]

    Tillich and G

    J.-P. Tillich and G. Z´ emor, Quantum ldpc codes with positive rate and minimum distance proportional to the square root of the blocklength, IEEE Transactions on Information Theory60, 1193 (2013)

    2013

  9. [17]

    Panteleev and G

    P. Panteleev and G. Kalachev, Quantum ldpc codes with almost linear minimum distance, IEEE Transactions on Information Theory68, 213 (2021)

    2021

  10. [18]

    N. P. Breuckmann and J. N. Eberhardt, Quantum low- density parity-check codes, PRX Quantum2, 040101 (2021)

    2021

  11. [19]

    P. W. Shor, Fault-tolerant quantum computation, inPro- ceedings of 37th conference on foundations of computer science(IEEE, 1996) pp. 56–65

    1996

  12. [20]

    Bomb ´ ın, Single-shot fault-tolerant quantum error cor- rection, Physical Review X5, 031043 (2015)

    H. Bomb ´ ın, Single-shot fault-tolerant quantum error cor- rection, Physical Review X5, 031043 (2015)

    2015

  13. [21]

    Leverrier, J.-P

    A. Leverrier, J.-P. Tillich, and G. Z´ emor, Quantum ex- pander codes, in2015 IEEE 56th Annual Symposium on Foundations of Computer Science(IEEE, 2015) pp. 810– 824

    2015

  14. [22]

    Panteleev and G

    P. Panteleev and G. Kalachev, Asymptotically good quantum and locally testable classical ldpc codes, inPro- ceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing(2022) pp. 375–388

    2022

  15. [23]

    Leverrier and G

    A. Leverrier and G. Z´ emor, Quantum tanner codes, arXiv preprint arXiv:2202.13641 (2022)

    2022 arXiv

  16. [24]

    Lin and M.-H

    T.-C. Lin and M.-H. Hsieh, Good quantum ldpc codes with linear time decoder from lossless expanders, arXiv preprint arXiv:2203.03581 (2022)

    2022 arXiv

  17. [25]

    Fawzi, A

    O. Fawzi, A. Grospellier, and A. Leverrier, Constant overhead quantum fault tolerance with quantum ex- pander codes, Communications of the ACM64, 106 (2020)

    2020

  18. [26]

    S. Gu, E. Tang, L. Caha, S. H. Choe, Z. He, and A. Ku- bica, Single-shot decoding of good quantum ldpc codes, arXiv preprint arXiv:2306.12470 (2023)

    2023 arXiv

  19. [27]

    Ashikhmin, C.-Y

    A. Ashikhmin, C.-Y. Lai, and T. A. Brun, Quantum data-syndrome codes, IEEE Journal on Selected Areas in Communications38, 449 (2020)

    2020

  20. [28]

    Delfosse, B

    N. Delfosse, B. W. Reichardt, and K. M. Svore, Beyond single-shot fault-tolerant quantum error correction, IEEE Transactions on Information Theory68, 287 (2021)

    2021

  21. [29]

    E. T. Campbell, A theory of single-shot error correction for adversarial noise, Quantum Science and Technology 4, 025006 (2019)

    2019

  22. [30]

    M. H. Freedman and M. B. Hastings, Quantum systems on non-k-hyperfinite complexes: A generalization of clas- sical statistical mechanics on expander graphs (2013), arXiv:1301.1363 [quant-ph]

    2013 arXiv

  23. [31]

    A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Physical Review A54, 1098 (1996)

    1996

  24. [32]

    A. M. Steane, Active stabilization, quantum computa- tion, and quantum state synthesis, Physical Review Let- ters78, 2252 (1997)

    1997

  25. [33]

    Bravyi and M

    S. Bravyi and M. B. Hastings, Homological product codes, inProceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC ’14 (Asso- ciation for Computing Machinery, New York, NY, USA,

  26. [34]

    M. B. Hastings, Quantum codes from high-dimensional manifolds (2016), arXiv:1608.05089 [quant-ph]

    2016 arXiv

  27. [35]

    Hatcher,Algebraic Topology(Cambridge University Press, Cambridge, 2002)

    A. Hatcher,Algebraic Topology(Cambridge University Press, Cambridge, 2002)

    2002

  28. [36]

    Elias, Error-free coding, Transactions of the IRE Pro- fessional Group on Information Theory4, 29 (1954)

    P. Elias, Error-free coding, Transactions of the IRE Pro- fessional Group on Information Theory4, 29 (1954)

    1954

  29. [37]

    MacWilliams and N

    F. MacWilliams and N. Sloane,The Theory of Error- Correcting Codes North-Holland(1977)

    1977

  30. [38]

    D. K. Tuckett, S. D. Bartlett, and S. T. Flammia, Ultra- high error threshold for surface codes with biased noise, Physical review letters120, 050505 (2018). 20

    2018

  31. [39]

    D. K. Tuckett, A. S. Darmawan, C. T. Chubb, S. Bravyi, S. D. Bartlett, and S. T. Flammia, Tailoring surface codes for highly biased noise, Physical Review X9, 041031 (2019)

    2019

  32. [40]

    Knill, R

    E. Knill, R. Laflamme, and W. H. Zurek, Resilient quan- tum computation, Science279, 342 (1998)

    1998

  33. [41]

    Aharonov and L

    D. Aharonov and L. Eldar, Quantum locally testable codes, SIAM Journal on Computing44, 1230 (2015)

    2015

  34. [42]

    A. Y. Kitaev, Quantum computations: algorithms and error correction, Russian Mathematical Surveys52, 1191 (1997)

    1997

  35. [43]

    B. J. Brown, N. H. Nickerson, and D. E. Browne, Fault- tolerant error correction with the gauge color code, Na- ture communications7, 12302 (2016)

    2016

  36. [44]

    Duivenvoorden, N

    K. Duivenvoorden, N. P. Breuckmann, and B. M. Terhal, Renormalization group decoder for a four-dimensional toric code, IEEE Transactions on Information Theory65, 2545 (2018)

    2018

  37. [45]

    A. O. Quintavalle, M. Vasmer, J. Roffe, and E. T. Camp- bell, Single-shot error correction of three-dimensional homological product codes, PRX Quantum2, 020340 (2021)

    2021

  38. [46]

    A. M. Kubica,The ABCs of the color code: A study of topological quantum codes as toy models for fault-tolerant quantum computation and quantum phases of matter, Ph.D. thesis, California Institute of Technology (2018)

    2018

  39. [47]

    Vasmer, D

    M. Vasmer, D. E. Browne, and A. Kubica, Cellular au- tomaton decoders for topological quantum codes with noisy measurements and beyond, Scientific reports11, 2027 (2021)

    2021

  40. [48]

    Fawzi, A

    O. Fawzi, A. Grospellier, and A. Leverrier, Efficient de- coding of random errors for quantum expander codes, in Proceedings of the 50th Annual ACM SIGACT Sympo- sium on Theory of Computing(2018) pp. 521–534

    2018

  41. [49]

    Grospellier and A

    A. Grospellier and A. Krishna, Numerical study of hy- pergraph product codes, arXiv preprint arXiv:1810.03681 (2018)

    2018 arXiv

  42. [50]

    N. P. Breuckmann and V. Londe, Single-shot decod- ing of linear rate ldpc quantum codes with high perfor- mance, IEEE Transactions on Information Theory68, 272 (2021)

    2021

  43. [51]

    Grospellier, L

    A. Grospellier, L. Grou` es, A. Krishna, and A. Leverrier, Combining hard and soft decoders for hypergraph prod- uct codes, Quantum5, 432 (2021)

    2021

  44. [52]

    Jack, Paths, trees, and flowers, Canadian Journal of mathematics17, 449 (1965)

    E. Jack, Paths, trees, and flowers, Canadian Journal of mathematics17, 449 (1965)

    1965

  45. [53]

    Panteleev and G

    P. Panteleev and G. Kalachev, Degenerate quantum ldpc codes with good finite length performance, Quantum5, 585 (2021)

    2021

  46. [54]

    Roffe, D

    J. Roffe, D. White, S. Burton, and E. Campbell, De- coding across the quantum ldpc code landscape (2020), arXiv preprint arXiv:2005.07016

    2020 arXiv

  47. [55]

    Sipser and D

    M. Sipser and D. A. Spielman, Expander codes, IEEE transactions on Information Theory42, 1710 (1996)

    1996

  48. [56]

    Scruby and K

    T. Scruby and K. Nemoto, Local probabilistic decod- ing of a quantum code, arXiv preprint arXiv:2212.06985 (2022)

    2022 arXiv

  49. [57]

    Hatcher,Algebraic Topology(Cambridge University Press, Cornell University, New York, 2001) available in Paperback

    A. Hatcher,Algebraic Topology(Cambridge University Press, Cornell University, New York, 2001) available in Paperback

    2001

  50. [58]

    D. P. Varodayan,Investigation of the Elias Product Code Construction for the Binary Erasure Channel, Tech. Rep. (2002)

    2002

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