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REVIEW 2 major objections 5 minor 62 references

Romanesco codes: Bias-tailored qLDPC codes from fractal codes

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Quantum code breaks the 2D distance barrier under biased noise.

desk verdict Romanesco codes: a genuinely new biased-noise qLDPC construction whose headline linear-distance claim is plausible but rests on empirical extrapolation from small tori. read the letter →

arxiv 2506.00130 v1 pith:2JLEVKP5 submitted 2025-05-30 quant-ph

Catherine Leroux , Joseph K. Iverson This is my paper
classification quant-ph PACS 03.67.Pp
keywords quantumLDPCcodesbiasednoisebiascellularautomatonfractalbivariatebicycleClifforddeformationerrorcorrection
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 introduces Romanesco codes, a family of quantum low-density parity-check (qLDPC) codes built from two classical cellular automaton codes and tailored to noise that is biased toward phase-flip errors. The central claim is that in the limit of infinitely strong bias the code splits into two independent classical codes, so the effective quantum distance approaches the classical distance of those inputs, which can grow linearly with the number of data qubits. That would beat the square-root distance scaling of two-dimensional topological codes while keeping local, low-weight stabilizers. Concretely, several reported families have an overhead factor $v_\infty = k d_c / n$ greater than 1, up to 2.67, meaning one patch encodes as much protection as several repetition or XZZX patches of the same size. Numerical simulations under code-capacity noise show strong suppression of the logical error rate compared with standard bivariate bicycle, XZZX, and thin surface codes.

What carries the argument

The central object is the Clifford-deformed bivariate bicycle code built from two cellular automaton input codes. A cellular automaton code is a classical LDPC code whose stabilizer generators are translation-invariant parity checks shaped by a binary rule matrix $R$; sweeping $R$ across the lattice generates the code. The paper reads $R_1$ and $R_2$ (a 180-degree rotation of $R_1$) as monomials in the cyclic shift variables $x$ and $y$ to form polynomials $A$ and $B = A^T$, producing a self-dual bivariate bicycle code. A local Hadamard (Clifford) rotation on one sublattice converts the CSS code into a non-CSS code whose stabilizers are half X and half Z. The load-bearing mechanism is the decoupling: because the X parts of the stabilizers live on disjoint sublattices, in the infinite-bias limit the decoding graph splits into two independent classical cellular automaton codes, so the lowest-weight all-Z logical operators have length $d_c$ and the effective quantum distance approaches $d_c$.

What would settle it

Compute the exact classical distance $d_c$ for a larger instance of the $[[N,12,D_{1,1}]]$ family, for example $H = L = 24$ with $N = 1152$ data qubits, using an exact integer-program decoder; if $d_c$ is not $3N/16$ or the ratio $d_c/n$ decays with size, the claimed linear scaling and the $v_\infty > 1$ overhead advantage fail.

Watch

Extended reading notes

Core claim

A Romanesco code is obtained by taking two cellular automaton (fractal) codes whose parity-check rules $R_1$ and $R_2$ are related by a 180-degree rotation, lifting them into a self-dual bivariate bicycle code with polynomials $A$ and $B = A^T$, and then applying a local Hadamard rotation on one of the two sublattices. The resulting code is non-CSS: every stabilizer generator is half X-type and half Z-type. Under strongly biased noise, the Z-type checks can be neglected, and the decoding graph separates into two decoupled sublattices, each supporting one of the input classical codes; the effective distance therefore approaches the classical distance $d_c$. For the families reported in Table I, $d_c$ scales linearly with the number $n$ of data qubits, for instance $d_c = 3N/16$ for the $[[N,12,D_{1,1}]]$ family, giving $v_\infty = k d_c / n > 1$ and up to 2.67, in contrast to repetition and twisted XZZX codes for which $v_\infty = 1$ and to 2D topological codes whose distance is bounded by $\sqrt{n}$. The paper further shows that on tori, cylinders, and open planes the classical distance remains linear, and that code-capacity simulations of the $[[288,12,12]]$ and $[[244,4,12]]$ codes outperform comparable bivariate bicycle and surface codes at moderate to large bias.

Load-bearing premise

The central claim rests on an empirical extrapolation: the classical distance of the cellular automaton input codes is inferred to grow linearly with lattice size from simulations on lattices up to about 1000 qubits, and no proof guarantees that larger lattices or other cellular automaton rules keep that growth.

Editorial extensions

If this is right

  • In the infinite bias limit, the effective distance equals the classical distance $d_c$, and for the Table I families $d_c$ grows linearly with $n$, giving $v_\infty = k d_c / n > 1$ up to 2.67.
  • A single Romanesco patch can replace several repetition or twisted XZZX patches of the same size for biased-noise quantum memory.
  • The $[[288,12,12]]$ code with $d_c = 54$ outperforms the $[[288,12,18]]$ bivariate bicycle code (infinite-bias distance 18) at moderate-to-large bias despite having a lower nominal quantum distance.
  • With open boundary conditions on a two-dimensional plane, $d_c$ still grows with $n$ (for example, $d_c > n/8$ for distance multiples of 12), unlike rotated XZZX codes whose infinite-bias distance scales as $\sqrt{n}$.
  • The codes retain self-duality and limited-range weight-8 stabilizers, giving transversal H, S, and CNOT gates and a bipartite honeycomb layout amenable to local hardware.

Reading between the lines

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

  • If the linear $d_c$ scaling holds asymptotically, biased-noise hardware such as cat qubits could obtain qLDPC-level overhead reductions without long-range connectivity, since the layout is two-dimensionally local; this is an extrapolation the paper does not fully prove.
  • The fractal-like structure of logical representatives suggests that cellular automaton decoders, which the paper mentions as future work, could provide efficient decoding at large bias.
  • Different boundary terminations might preserve more logical qubits; the paper's simple truncation reduces $k$ by half on cylinders and to 4 on the plane, so boundary engineering is a natural lever for improving $v_\infty$.
  • A direct proof or disproof of $d_c \sim n$ for the Table I families would settle whether the advantage over 2D topological codes persists asymptotically.
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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

2 major / 5 minor

Summary. The paper introduces Romanesco codes, a family of Clifford-deformed bivariate bicycle codes whose input classical codes are cellular automaton (fractal) codes. The construction guarantees that in the infinite-noise-bias limit the decoding graph decouples into two independent classical cellular automaton codes, so the effective code distance approaches the classical distance dc. The authors numerically search over cellular automaton rules and identify several families of codes on the torus, cylinder, and open-boundary 2D plane. They report code families with encoding rates up to k=16 and overhead factors v∞ = kdc/n up to 2.67 on the torus, and they present code-capacity Monte Carlo simulations showing improved logical error rates compared with bivariate bicycle codes and rotated surface codes under biased noise. The paper is well structured and includes detailed appendices describing the search, decoding algorithms, and boundary constructions.

Significance. If the claimed asymptotic distance scalings are correct, the Romanesco construction provides a concrete route to qLDPC codes on a 2D local lattice whose infinite-bias effective distance grows linearly with n, surpassing the O(√n) distance limit of 2D topological codes under strong bias. The reduction to two decoupled classical cellular automaton codes is elegant and directly connects the quantum code performance to the well-studied distance properties of classical LDPC codes. The numerical results show substantial practical improvements under biased noise, and the paper is careful to discuss the regime of validity of the effective-distance picture. The main weakness is that the central asymptotic claim rests on empirical extrapolations from lattices with fewer than 1000 qubits, with irregular open-boundary data and no proof of linear scaling.

major comments (2)
  1. [Section III.B and Table I] The claimed linear scalings of dc with n are presented as facts for the families in Table I, but they are inferred from lattices with fewer than 1000 qubits, and no finite-size data are shown for the torus or cylinder families. The open-boundary data in Table VII and Fig. 8 are highly irregular: for the k=2 family, dc/n equals 0.20, 0.11, 0.12, and 0.27 at d = 17, 18, 19, and 23, respectively, and for the k=4 family it drops from 0.16 at d=12 to 0.08 at d=16 and 0.04 at d=20 before returning to 0.15 at d=24. The statement in Section III.D that "the infinite bias distance dc scales linearly with the number of data qubits n" is therefore not supported by the evidence provided. Since v∞ = kdc/n > 1 is the central quantitative claim, the authors should either prove the linear scaling for the identified CA-rule families or provide complete finite-size tables/plots for each family and explicitly label the asymptotic scaling as an empirical conjecture.
  2. [Appendix B, Algorithm 1] The distances dc for the torus and cylinder families are computed with Algorithm 1, which the authors state returns an upper bound on the code distance. The convergence check (generating 100 new low-weight logicals) is heuristic, and if the bound is not tight at the largest searched sizes, the reported linear scalings and the corresponding v∞ values in Table I would be overestimates. The authors should run an exact distance computation for at least the largest instances of each torus family, as was done for the open-boundary codes in Table VII, and state explicitly whether the bounds are tight. Without this, the headline advantage over 2D topological codes is not firmly established.
minor comments (5)
  1. [Table I] The constraints such as H%3=0 and L%3=0 are used without defining the modulo notation in the caption; please add a brief explanation.
  2. [Section III.B] The sentence "We use this geometric constraint to find the most promising stabilizer shapes" refers to the 9q^2 square-lattice search constraint; clarify that this is a search heuristic and not a requirement of the final code families.
  3. [Fig. 2 caption] The caption states that shaded regions are 95% confidence intervals for a Poisson distribution, but does not report the number of Monte Carlo shots or the number of logical errors per point; please include this information.
  4. [Appendix E, Fig. 8] The phrase "the classical distance does grow with 2d^2 > n" is confusing; it should be rephrased as "dc grows as Θ(d^2), which is proportional to n" or similar.
  5. [Section IV] The term "beam climbing" in the description of the tesseract decoder is not standard and is not defined; please clarify whether this is a specific feature of the tesseract decoder and cite the relevant description.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the infinite-bias reduction to cellular automaton codes is structural, and the Table I scalings are honest empirical extrapolations rather than fitted predictions.

full rationale

The paper's central derivation is not circular. The claim that, in the infinite-bias limit, a Romanesco code reduces to two independent classical cellular automaton codes follows directly from the code construction: after the Hadamard rotation, the X-type checks on the black and gray sublattices are supported on separate sectors with no cross terms, as shown in Eq. (1). The effective distance then approaching the classical distance dc is a structural consequence of the parity-check block form, not an input assumed to prove the conclusion. The values of d and dc are computed by an explicit distance-finding algorithm (Algorithm 1) on the parity-check matrices, and the linear scalings in Table I are explicitly described as empirical findings from multiple lattice sizes rather than as fitted parameters renamed as predictions. The numerical search over cellular automaton rules is an optimization over an independently specified rule space, and the logical-error-rate simulations are benchmarked against external codes such as the XZZX surface code and the bivariate bicycle code. The self-citations present (e.g., refs. [3], [7], [9], [14]) are contextual citations to cat-qubit hardware and prior experimental work, and none of them carries the load of the distance-scaling or effective-distance argument. The open-boundary irregularity of dc noted in Table VII is a limitation of the empirical extrapolation, but it is a correctness-risk concern, not circularity. Overall, no step in the claimed derivation reduces by definition to its own inputs.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The construction is explicit and depends only on standard properties of cyclic shifts and on the biased-noise model; the selected CA rules are design variables, and no physical entity is invented.

free parameters (2)
  • Cellular automaton rule pair (R1, R2) = bowtie and butterfly shapes in figs. 6/7, table I
    The rule shapes are selected by a numerical search over m=2,3,4 and w=3,4 on finite lattices to maximize v_infinity = k dc / n; all code parameters and the reported distance scalings depend on this design choice.
  • Family lattice constraints and fixed k = H percent 3 = 0, L percent 3 = 0, etc. in table I
    These divisibility constraints are introduced after the search to group instances that happen to share the same k and dc(n); they are not derived from a general theorem.
assumptions (4)
  • standard math Cyclic shift matrices x and y commute, so H_X H_Z^T = 0 for any bivariate bicycle code defined by polynomials A and B.
    Used in Section II.A to guarantee that the CSS code C has commuting stabilizers before the Clifford deformation.
  • domain assumption In the pZ >> pX limit, Z errors dominate and the X-type parts of the mixed stabilizers on each sublattice form two independent decoding graphs.
    Justifies the reduction to two classical CA codes in Section III.A and the effective distance claim; relies on the biased noise model.
  • domain assumption The selected classical CA codes from [13] have distance that can grow linearly with blocklength.
    The paper leans on [13] for the quality of the input codes; the specific searched rules are assumed to inherit large distance.
  • ad hoc to paper For open boundary conditions, truncating stabilizers and adding corner qubits preserves commutation for the bowtie shape.
    Section III.D and appendix E verify commutation only for two specific shapes; no general boundary prescription is proven.

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Pith. "Pith review of Romanesco codes: Bias-tailored qLDPC codes from fractal codes." pith.science (2026). https://pith.science/paper/2JLEVKP5

@misc{pith2026250600130,
  author       = {Pith},
  title        = {Pith review of: Romanesco codes: Bias-tailored qLDPC codes from fractal codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2JLEVKP5}},
  note         = {Machine review of arXiv:2506.00130}
}
read the original abstract

We introduce and analyze a family of Clifford-deformed bivariate bicycle codes that are tailored for biased noise. Our qLDPC codes are defined on a bipartite hexagonal lattice with limited-range gates and low-weight stabilizers. The code is non-CSS, featuring stabilizer generators that are each half X and half Z. We find small examples with high encoding rate that perform well for a large range of bias. In the limit of large noise bias, the code reduces to two independent classical cellular automaton codes, giving a distance scaling better than is possible with 2D topological quantum codes. Our construction combines two classical cellular automaton codes, LDPC codes that were recently proposed for use with noise-biased cat qubits, related to each other by a reflection. Each stabilizer in the quantum code is obtained by multiplying an all-X stabilizer from the first code with an all-Z stabilizer from the second code, or the other way around. The result is a self-dual quantum code with a number of qubits equal to the sum of the input codes and stabilizer weight and locality determined by the input codes. Under strong noise bias, the effective distance of the quantum code approaches the distance of the input codes. We simulate the logical performance of our qLDPC codes under code-capacity noise and find strong suppression of the logical error rate.

Figures

Figures reproduced from arXiv: 2506.00130 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic representation of our code construction. 1. We specify the stabilizer shape of the cellular [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Total logical error rate (average of the X and Z [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The XZ [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (6 more)
Figure 5
Figure 5. Figure 5: FIG. 5. The first two lowest-weight mixed logical op [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. a) Cellular automaton rules [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. a) Cellular automaton rules [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Classical distance as a function of the quantum [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 10
Figure 10. Figure 10: FIG. 10. The fraction of shots where the tesseract de [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Logical error rate of (a) [[288 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

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Reference graph

Works this paper leans on

62 extracted references · 45 canonical work pages

  1. [1]

    Sweep rule in each sublattice 3

    Specify the rules 2. Sweep rule in each sublattice 3. Interleave the sublattices X X X X X XX X Z Z Z Z Z ZZ Z S1 S2

    work page

  2. [2]

    These errors have linear sensitivity in the noise bias, meaning that their performance reduces to that of two decoupled classical LDPC codes

    Searching for quantum codes We do not allow for d = 2 codes: these cases directly imply that there exists a weight-2 mixed logical operator with one Pauli Z and one Pauli X that connects a black qubit to a gray qubit in C♣ since the classical code in each sub-lattice strictly have higher weight logical operators. These errors have linear sensitivity in th...

    work page

  3. [3]

    Like on the torus and cylin- der, the all-Z logical representatives are supported by the cellular automaton code on each sub-lattice

    We list instances of Romanesco codes on a two- dimensional plane in table IV with equal height and width, since the quantum distance is given by the minimum dimension. Like on the torus and cylin- der, the all-Z logical representatives are supported by the cellular automaton code on each sub-lattice. We find that the number of logical qubits in table IV i...

    work page

  4. [4]

    Choose layout X X X X Z ZZ Z S1 X XX X Z Z Z Z S2 FIG. 1. Schematic representation of our code construction. 1. We specify the stabilizer shape of the cellular automaton code on each sub-lattice, black or gray, as a binary matrix labelled R1 or R2. Here R2 is related to R1 by a 180◦ rotation so that all stabilizers commute. 2. We build the parity check ma...

    work page

  5. [5]

    We also make sure that the bottom row and left column have at least a non-zero entry

    Searching for classical codes We generate all unique cellular automaton rules R of size m × m with weight w in both rows and columns (taking the periodic boundary conditions into account). We also make sure that the bottom row and left column have at least a non-zero entry. For each cellular automaton rule, we generate the parity check matrix with periodi...

    work page

  6. [6]

    Our approach is as follows

    Selecting the best codes Next we need to define what are good code pa- rameters for this code construction. Our approach is as follows. For each unique k value we obtained for quantum codes, we keep the quantum codes with the best classical code distance (or infinite bias dis- tance) d∞. Among these codes, we then keep those with the minimum stabilizer we...

    work page

  7. [7]

    We found codes with interesting parameters in table V

    Code families In order to find the most promising stabilizer shapes we constrained the honeycomb lattice to have the same height and width. We found codes with interesting parameters in table V. Here we vary the Algorithm 2. Pseudo-code to build a Romanesco code on a bipartite honeycomb lattice with periodic boundary conditions with 2HL qubits: each sub-l...

    work page

  8. [8]

    bowtie” codes and those with the bulk stabilizer in fig. 7 a) as “butter- fly

    Stabilizer generators In figs. 6 and 7 we show the stabilizers for the code families in table I on the torus, the cylinder and two- dimensional plane. We will refer to the codes with the bulk stabilizer in fig. 6 a) as “bowtie” codes and those with the bulk stabilizer in fig. 7 a) as “butter- fly” codes. We were only able to find bowtie codes on the two-d...

    work page

Show all 62 references
  1. [9]

    The code families on the cylinder are also shown in table III

    Code families The code families on the torus are presented in table I. The code families on the cylinder are also shown in table III. As highlighted in the main text, the code families on the cylinder have parameters that are similar to those on the torus, with the ex- ception...

  2. [10]

    We sample the representatives like we did on the torus (see appendix C)

    Logical representatives Here we investigate the minimum-weight mixed logical representatives on the two-dimensional plane. We sample the representatives like we did on the torus (see appendix C). The distribution is shown in table VIII for the small to moderate size codes in t...

  3. [11]

    P. T. Cochrane, G. J. Milburn, and W. J. Munro, Macroscopically distinct quantum-superposition states as a bosonic code for amplitude damping, Phys. Rev. A 59, 2631 (1999)

    1999

  4. [12]

    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 Physics 16, 509 (2020)

    2020

  5. [13]

    Putterman, K

    H. Putterman, K. Noh, R. N. Patel, G. A. Peairs, G. S. MacCabe, M. Lee, S. Aghaeimeibodi, C. T. Hann, I. Jarrige, G. Marcaud, Y. He, H. Moradine- jad, J. C. Owens, T. Scaffidi, P. Arrangoiz-Arriola, J. Iverson, H. Levine, F. G. S. L. Brand˜ ao, M. H. Matheny, and O. Painter, P...

    2024 arXiv

  6. [14]

    Aliferis and J

    P. Aliferis and J. Preskill, Fault-tolerant quantum computation against biased noise, Phys. Rev. A 78, 052331 (2008)

    2008

  7. [15]

    Aliferis, F

    P. Aliferis, F. Brito, D. P. DiVincenzo, J. Preskill, M. Steffen, and B. M. Terhal, Fault-tolerant com- puting with biased-noise superconducting qubits: a case study, New Journal of Physics 11, 013061 (2009)

    2009

  8. [16]

    D. K. Tuckett, S. D. Bartlett, and S. T. Flammia, Ultrahigh error threshold for surface codes with bi- ased noise, Phys. Rev. Lett. 120, 050505 (2018)

    2018

  9. [17]

    Chamberland, K

    C. Chamberland, K. Noh, P. Arrangoiz-Arriola, E. T. Campbell, C. T. Hann, J. Iverson, H. Putter- man, T. C. Bohdanowicz, S. T. Flammia, A. Keller, G. Refael, J. Preskill, L. Jiang, A. H. Safavi-Naeini, O. Painter, and F. G. Brand˜ ao, Building a fault- tolerant quantum compute...

    2022

  10. [18]

    F.-M. L. R´ egent, C. Berdou, Z. Leghtas, J. Guillaud, and M. Mirrahimi, High-performance repetition cat code using fast noisy operations, Quantum 7, 1198 (2023)

    2023

  11. [19]

    Putterman, K

    H. Putterman, K. Noh, C. T. Hann, G. S. Mac- Cabe, S. Aghaeimeibodi, R. N. Patel, M. Lee, W. M. Jones, H. Moradinejad, R. Rodriguez, N. Mahuli, J. Rose, J. C. Owens, H. Levine, E. Rosenfeld, P. Reinhold, L. Moncelsi, J. A. Alcid, N. Ali- doust, P. Arrangoiz-Arriola, J. Barnett...

    2025

  12. [20]

    Gouzien, D

    E. Gouzien, D. Ruiz, F.-M. Le R´ egent, J. Guillaud, and N. Sangouard, Performance analysis of a repe- tition cat code architecture: Computing 256-bit el- liptic curve logarithm in 9 hours with 126 133 cat qubits, Phys. Rev. Lett. 131, 040602 (2023)

    2023

  13. [21]

    Guillaud and M

    J. Guillaud and M. Mirrahimi, Repetition cat qubits for fault-tolerant quantum computation, Phys. Rev. X 9, 041053 (2019)

    2019

  14. [22]

    Guillaud and M

    J. Guillaud and M. Mirrahimi, Error rates and re- source overheads of repetition cat qubits, Phys. Rev. A 103, 042413 (2021)

    2021

  15. [23]

    D. Ruiz, J. Guillaud, A. Leverrier, M. Mirrahimi, and C. Vuillot, Ldpc-cat codes for low-overhead quantum computing in 2d, Nature Communications 16, 1040 (2025)

    2025

  16. [24]

    C. T. Hann, K. Noh, H. Putterman, M. H. Ma- theny, J. K. Iverson, M. T. Fang, C. Cham- berland, O. Painter, and F. G. S. L. Brand˜ ao, Hybrid cat-transmon architecture for scalable, hardware-efficient quantum error correction (2024), arXiv:2410.23363 [quant-ph]

    2024

  17. [25]

    J. P. Bonilla Ataides, D. K. Tuckett, S. D. Bartlett, S. T. Flammia, and B. J. Brown, The xzzx surface code, Nature Communications 12, 2172 (2021)

    2021

  18. [26]

    A. S. Darmawan, B. J. Brown, A. L. Grimsmo, D. K. Tuckett, and S. Puri, Practical quantum error cor- rection with the xzzx code and kerr-cat qubits, PRX Quantum 2, 030345 (2021)

    2021

  19. [27]

    D. K. Tuckett, A. S. Darmawan, C. T. Chubb, S. Bravyi, S. D. Bartlett, and S. T. Flammia, Tai- loring surface codes for highly biased noise, Phys. Rev. X 9, 041031 (2019)

    2019

  20. [28]

    Srivastava, A

    B. Srivastava, A. Frisk Kockum, and M. Granath, The XYZ 2 hexagonal stabilizer code, Quantum 6, 698 (2022)

    2022

  21. [29]

    Liang, F

    Z. Liang, F. Yang, Z. Yi, and X. Wang, Quan- tum xyz cyclic codes for biased noise (2025), arXiv:2501.16827 [quant-ph]

    2025 arXiv

  22. [30]

    Q. Xu, N. Mannucci, A. Seif, A. Kubica, S. T. Flam- mia, and L. Jiang, Tailored xzzx codes for biased noise (2022), arXiv:2203.16486 [quant-ph]

    2022

  23. [31]

    J. F. San Miguel, D. J. Williamson, and B. J. Brown, A cellular automaton decoder for a noise-bias tai- lored color code, Quantum 7, 940 (2023)

    2023

  24. [32]

    J. A. Campos and K. R. Brown, Clifford-deformed 20 compass codes (2024), arXiv:2412.03808 [quant-ph]

    2024 arXiv

  25. [33]

    A. Dua, A. Kubica, L. Jiang, S. T. Flammia, and M. J. Gullans, Clifford-deformed surface codes, PRX Quantum 5, 010347 (2024)

    2024

  26. [34]

    Setiawan and C

    F. Setiawan and C. McLauchlan, Tailoring dynam- ical codes for biased noise: The x 3z3 floquet code (2024), arXiv:2411.04974 [quant-ph]

    2024

  27. [35]

    A. A. Kovalev and L. P. Pryadko, Improved quan- tum hypergraph-product ldpc codes, in 2012 IEEE International Symposium on Information Theory Proceedings (2012) pp. 348–352

    2012

  28. [36]

    Tillich and G

    J.-P. Tillich and G. Z´ emor, Quantum ldpc codes with positive rate and minimum distance pro- portional to the square root of the blocklength, IEEE Transactions on Information Theory 60, 1193 (2014)

    2014

  29. [37]

    N. P. Breuckmann and J. N. Eberhardt, Quantum low-density parity-check codes, PRX Quantum 2, 040101 (2021)

    2021

  30. [38]

    N. P. Breuckmann and J. N. Eberhardt, Balanced product quantum codes, IEEE Transactions on In- formation Theory 67, 6653 (2021)

    2021

  31. [39]

    M. B. Hastings, J. Haah, and R. O’Donnell, Fiber bundle codes: breaking the n1/2 polylog(n) bar- rier for quantum ldpc codes, in Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021 (Association for Com- puting Machinery, New York, NY, USA, 202...

    2021

  32. [40]

    Panteleev and G

    P. Panteleev and G. Kalachev, Quantum ldpc codes with almost linear minimum distance, IEEE Trans- actions on Information Theory 68, 213 (2022)

    2022

  33. [41]

    Panteleev and G

    P. Panteleev and G. Kalachev, Asymptotically good quantum and locally testable classical ldpc codes, in Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022 (Association for Computing Machinery, New York, NY, USA, 2022) p. 375–388

    2022

  34. [42]

    A. A. Kovalev and L. P. Pryadko, Quantum kro- necker sum-product low-density parity-check codes with finite rate, Physical Review A—Atomic, Molec- ular, and Optical Physics 88, 012311 (2013)

    2013

  35. [43]

    Bravyi, A

    S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, High-threshold and low- overhead fault-tolerant quantum memory, Nature 627, 778 (2024)

    2024

  36. [44]

    Roffe, L

    J. Roffe, L. Z. Cohen, A. O. Quintavalle, D. Chan- dra, and E. T. Campbell, Bias-tailored quantum LDPC codes, Quantum 7, 1005 (2023)

    2023

  37. [45]

    E. D. de Carvalho, W. S. Soares, and E. B. da Silva, Topological quantum codes from lattices partition on the n-dimensional flat tori, Entropy 23, 10.3390/e23080959 (2021)

    2021 doi

  38. [46]

    Kubica and M

    A. Kubica and M. E. Beverland, Universal transver- sal gates with color codes: A simplified approach, Phys. Rev. A 91, 032330 (2015)

    2015

  39. [47]

    Delfosse, M

    N. Delfosse, M. E. Beverland, and M. A. Tremblay, Bounds on stabilizer measurement circuits and ob- structions to local implementations of quantum ldpc codes (2021), arXiv:2109.14599 [quant-ph]

    2021 arXiv

  40. [48]

    Baspin and A

    N. Baspin and A. Krishna, Connectivity constrains quantum codes, Quantum 6, 711 (2022)

    2022

  41. [49]

    Baspin, V

    N. Baspin, V. Guruswami, A. Krishna, and R. Li, Improved rate-distance trade-offs for quan- tum codes with restricted connectivity (2023), arXiv:2307.03283 [quant-ph]

    2023 arXiv

  42. [50]

    Roffe, D

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

    2020

  43. [51]

    Roffe, LDPC: Python tools for low density parity check codes (2022)

    J. Roffe, LDPC: Python tools for low density parity check codes (2022)

    2022

  44. [52]

    M. S. Kesselring, F. Pastawski, J. Eisert, and B. J. Brown, The boundaries and twist defects of the color code and their applications to topological quantum computation, Quantum 2, 101 (2018)

    2018

  45. [53]

    Thomsen, M

    F. Thomsen, M. S. Kesselring, S. D. Bartlett, and B. J. Brown, Low-overhead quantum computing with the color code, Phys. Rev. Res. 6, 043125 (2024)

    2024

  46. [54]

    Steffan, S

    V. Steffan, S. H. Choe, N. P. Breuckmann, F. R. F. Pereira, and J. N. Eberhardt, Tile codes: High- efficiency quantum codes on a lattice with boundary (2025), arXiv:2504.09171 [quant-ph]

    2025 arXiv

  47. [55]

    Liang, J

    Z. Liang, J. N. Eberhardt, and Y.-A. Chen, Planar quantum low-density parity-check codes with open boundaries (2025), arXiv:2504.08887 [quant-ph]

    2025 arXiv

  48. [56]

    L. A. Beni, O. Higgott, and N. Shutty, Tesseract: A search-based decoder for quantum error correction (2025), arXiv:2503.10988 [quant-ph]

    2025 arXiv

  49. [57]

    Panteleev and G

    P. Panteleev and G. Kalachev, Degenerate Quan- tum LDPC Codes With Good Finite Length Per- formance, Quantum 5, 585 (2021)

    2021

  50. [58]

    Higgott and C

    O. Higgott and C. Gidney, Sparse Blossom: correct- ing a million errors per core second with minimum- weight matching, Quantum 9, 1600 (2025)

    2025

  51. [59]

    Virtanen, R

    P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, ˙I. Polat, Y. Feng...

    2020

  52. [60]

    Chandra, G

    O. Chandra, G. Muraleedharan, and G. K. Brennen, Non-local resources for error correction in quantum ldpc codes (2024), arXiv:2409.05818 [quant-ph]

    2024

  53. [61]

    Vasmer, D

    M. Vasmer, D. E. Browne, and A. Kubica, Cellular automaton decoders for topological quantum codes with noisy measurements and beyond, Scientific Re- ports 11, 2027 (2021)

    2021

  54. [62]

    Wolfram, Statistical mechanics of cellular au- tomata, Rev

    S. Wolfram, Statistical mechanics of cellular au- tomata, Rev. Mod. Phys. 55, 601 (1983)

    1983

Pith tools

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