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Demystifying the Balanced Product Code: A Review

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper shows that balanced product quantum codes reduce to a concrete pair of parity-check matrices built from an expanding matrix with a cyclic symmetry, and that these matrices yield low-density CSS codes with distance scaling…

desk verdict Solid review that earns its keep as a teaching tool; just be aware the headline distance scaling rests on an unproved (but cited) balancing lemma. read the letter →

arxiv 2505.13679 v2 pith:K7F4CBTW submitted 2025-05-19 quant-ph

Heather Leitch , Alastair Kay This is my paper · ORCID
classification quant-ph PACS 03.67.Pp
keywords balancedproductcodequantumLDPCcodesCSSexpandergraphsparity-checkmatricesdistancebalancingstoragedensityLPS
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to make balanced product quantum LDPC codes usable by rewriting the construction directly in terms of parity-check matrices, removing the chain-complex machinery of the original presentation. Its central claim is that when a binary matrix $I$ with an $(\alpha,\beta)$-expanding property carries an order-$l$ permutation symmetry ($R I = I C^T$ with $R^l = C^l = 1$), the two matrices $H_X = [I^T, 1+C]$ and $H_Z = [1+R, I]$ automatically define a CSS code: the commutation $H_X H_Z^T = 0$ holds, the code is low density, and its distances are linear in $l$ and $m$ respectively. The paper works through the distance proof, then adds a distance-balancing construction that equalizes the two distances, and applies the whole recipe to LPS expander graphs. If correct, this gives a concrete path from a classical expander graph to a quantum code with parameters $O([[N, N^{4/5}, N^{3/5}]])$, a storage density $kd^2$ on the order of $N^2$, and a distance that grows faster than $\sqrt{N}$.

What carries the argument

The central object is the balanced product parity-check pair, $H_X = [I^T, 1+C]$ and $H_Z = [1+R, I]$. The matrix $I$ is the incidence matrix of an expanding graph; the permutation matrices $R$ and $C$ implement an order-$l$ symmetry $R I = I C^T$, and the blocks $1+R$ and $1+C$ are parity-check matrices of repetition codes over each orbit. The symmetry does two jobs: it ensures the CSS orthogonality $H_X H_Z^T = 0$ modulo 2, and it is the mechanism that quotients out the orbit structure, removing the tensor-product blow-up in qubit count that limits hypergraph products. The distance argument then runs on expansion: any short dressed logical operator would produce a short vector whose image under $I$ or $I^T$ expands beyond the allowed weight. A secondary mechanism is Definition 3's distance balancing, a block construction combining the quantum code with a good classical LDPC code to equalize $d_X$ and $d_Z$ at the cost of more physical qubits.

What would settle it

Fix the LPS incidence matrix for a growing sequence of primes $q$ (holding $p$ fixed), build $H_X$ and $H_Z$ from Definition 1, and compute the exact minimum weights of dressed logical operators; if the $X$-distance or $Z$-distance stops growing linearly in $l$ or $m$ with fixed positive constants, Theorem 1's distance claim is false. Then apply the same exact search to the distance-balanced matrices and look for any logical operator of weight below $d_X d_C$ or $d_Z$.

Watch

Extended reading notes

Core claim

The central claim is Definition 1 plus Theorem 1: for an $(\alpha,\beta)$-expanding matrix $I \in \{0,1\}^{m\times n}$ with row and column weight at most $s$, a cyclic symmetry $R I = I C^T$ whose orbits all have length $l$, and $\max(|\ker I|, |\ker I^T|) = k_0$, the parity-check matrices $H_X = [I^T, 1+C]$ and $H_Z = [1+R, I]$ define a low-density CSS code with parameters $[[n+m, k_0/l, d_X = \gamma_X l, d_Z = \gamma_Z m]]$. The distance proof shows that expansion forbids short logical operators: a dressed logical operator localized on the left block would force a vector $u(t_0)$ of weight below $\alpha m$ whose image under $I^T$ is too heavy, while the symmetric orbit-sum construction gives $X$-logicals of weight $l$. The paper further claims that Definition 3's distance balancing converts such a code into one with equalized distances, and that with the LPS expander family this yields $O([[N,N^{4/5},N^{3/5}]])$ quantum LDPC codes.

Load-bearing premise

The load-bearing premise is that a family of expanding matrices with exactly the cyclic symmetry of Definition 1 exists, together with the paper's explicit choice in Section III not to prove that distance balancing creates no dressed logical operators shorter than $d_X d_C$ or $d_Z$.

Editorial extensions

If this is right

  • Given any expanding matrix $I$ with the required cyclic symmetry, the recipe produces a quantum LDPC code immediately from its incidence matrix, with no chain-complex or fiber-bundle abstraction needed.
  • For LPS expander graphs with fixed $p$ and growing $q$, the balanced product gives a code whose parameters before balancing are $O(q^3)$ physical qubits, $O(q^2)$ logical qubits, and distances $O(q)$ and $O(q^3)$; after distance balancing this becomes $O([[N,N^{4/5},N^{3/5}]])$.
  • The construction reaches $kd^2 \sim N^2$, the same storage-density scaling as hypergraph product codes, but with distance scaling $N^{3/5}$ instead of $\sqrt{N}$, so the code's storage density is no worse while its distance grows strictly faster.
  • The paper's bound $k \geq k_0/l$ means the number of logical qubits could in principle be larger than $k_0/l$; in the explored examples the bound is saturated, so exceeding it would push past the $kd^2 \sim N^2$ threshold.
  • The distance-balancing step turns any asymmetric $[[n,k,d_X,d_Z]]$ code into an $O([[n\max(d_X,d_Z)/\min(d_X,d_Z), k\max(d_X,d_Z)/\min(d_X,d_Z), \max(d_X,d_Z)]])$ code, so the balancing recipe is reusable beyond balanced products.

Reading between the lines

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

  • A natural next search is for families of expanding matrices whose kernels contain vectors with far more symmetry than the $k_0/l$ orbit average guarantees; if such families exist, the lower bound $k \geq k_0/l$ could be beaten and balanced product codes would exceed, rather than match, the $kd^2 \sim N^2$ threshold.
  • Because the paper leaves the part of distance balancing that rules out shorter logical operators to the reader, the balancing claim can be stress-tested on finite examples by building the parity-check matrices explicitly and checking whether any dressed logical operator below $d_X d_C$ or $d_Z$ exists.
  • The Tanner code variant suggests a design freedom the paper does not develop: choosing local codes with growing distance $d_l$ could multiply the $X$-distance by $d_l$ while adding physical qubits, potentially producing a different trade-off curve between $N$, $k$, and $d$.
  • The small LPS example with parameters $[[360,26,5]]$ and its subsystem variant offers a concrete benchmark for recomputing exact distances and for probing how the constants $\gamma_X$ and $\gamma_Z$ behave as $q$ grows.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper is a pedagogical review of the balanced product quantum LDPC code construction of Breuckmann and Eberhardt. It presents the construction directly in terms of parity-check matrices: for an expanding binary matrix I with an order-l permutation symmetry, the CSS code is defined by H_X = [I^T, 1+C] and H_Z = [1+R, I] (Eq. (3)). The paper proves the CSS condition, counts logical qubits, gives a distance proof for the asymmetric code following Panteleev--Kalachev, describes a distance-balancing procedure from Evra--Kaufman--Zémor, and combines these with LPS expander graphs to claim codes with parameters O([[N,N^{4/5},N^{3/5}]]). Worked examples, including a 6-cycle repetition code and an LPS graph with p=3, q=5, are checked with a SAT solver whose code and data are deposited in two repositories.

Significance. If the exposition is accurate, the paper serves a useful purpose: it lowers the barrier to understanding a landmark construction in quantum LDPC code theory, provides explicit parity-check matrices rather than abstract chain complexes, and gives independently checkable examples and data. The claimed asymptotic O([[N,N^{4/5},N^{3/5}]]) is already established in the literature, so the paper's contribution is pedagogical rather than novel in results. The main strengths are the concrete matrix formulation, the worked examples with machine-checked distances, and the attempt to give a self-contained distance proof for the asymmetric code. However, the final asymptotic is only as strong as the distance-balancing lower bound, which the manuscript does not prove, and the main distance proof contains a gap in the X-distance part.

major comments (3)
  1. [Section III, Definition 3] The distance-balancing lower bounds \tilde d_X = d_X d_C and \tilde d_Z = d_Z are asserted immediately after Definition 3, but the text explicitly says "We will not give the full details here" and only constructs logical operators of weight d_X d_C and d_Z. The final claim in Section IV.A that the balanced product with LPS graphs yields O([[N,N^{4/5},N^{3/5}]]) is load-bearing on exactly this missing lower bound: the balanced distance is obtained by multiplying the O(q) X-distance by a classical code with d_C = O(q^2). As written, the manuscript does not rule out shorter dressed operators in the balanced code, so the headline scaling is not established by the review. Please either provide the full proof of the distance-balancing lower bounds or state the result as an imported theorem with a precise citation to [20], and explicitly mark in Section IV that the final scaling depends on this external result.
  2. [Section II.C, X-distance proof] In the X-distance proof, t0 is defined as the smallest integer such that |v(t0)| > (1−α)l/s, and the text derives |v(t0)| ≤ ((s−1)α+1)l/s. It then claims that choosing γ_X < β((s−1)α+1)/(2s) contradicts the expansion property. This does not follow: expansion only gives |Iv(t0)| ≥ β|v(t0)| > β(1−α)l/s, while the proof's upper bound is |Iv(t0)| ≤ 2|u| < 2γ_X l. To obtain a contradiction one needs 2γ_X l < β(1−α)l/s, which is not implied by the displayed choice of γ_X because ((s−1)α+1) > (1−α). As written, the proof does not establish the claimed d_X = γ_X l bound; the argument would need either the smaller choice γ_X < β(1−α)/(2s) or an additional argument showing that |v(t0)| is actually close to its upper bound.
  3. [Section IV] The asymptotic claim depends on the existence of a family of expanding matrices I with the exact order-l symmetry of Definition 1, bounded row and column weight s, and kernel dimension scaling as q^2; this is imported from the LPS graphs [22], and the manuscript states without proof that these graphs have "an appropriate symmetry of order l = q." Similarly, the random-expander claim in the first paragraph of Section IV ("With high probability, we get a code that is O([[nl,nl,l,n]]) before distance balancing") is asserted without proof. Since this is a review, importing established results is acceptable, but the provenance and precise statements of these existence claims should be made explicit in the text so that the final O([[N,N^{4/5},N^{3/5}]]) is clearly presented as a combination of cited results rather than a derivation completed in this paper.
minor comments (4)
  1. [Section II.C, footnote 30] The remark "Assuming a large enough code 2βγXl<αn that a token case can be neglected" is too vague for a proof that claims to give full details; the precise condition on n/l needed for the expansion property to apply to v(t0) should be stated in the main text.
  2. [Example 2] In the list of orbit elements, "AH, AH 2, AH 4, AH 4" appears to contain a typo; it should presumably be "AH, AH^2, AH^3, AH^4".
  3. [Section I] The dimension formula is written as k = n − mA − mB, but the symbols mA and mB are not defined at that point; the later text uses mX and mZ, so the notation should be harmonized.
  4. [Example 2] The sentence "The X and Z distances satisfy dX,dZ ≤ 5" followed by "Hence, the code is [[360,26,5]]" would be clearer if the text explicitly noted that the upper bounds are verified to be tight by the SAT solver.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the review proves or externalizes its load-bearing claims; self-cited tools are checkable, and the only gap is an explicit unproved lower bound, not a circular definition.

full rationale

The paper is a review that re-presents the Breuckmann–Eberhardt balanced product construction [1] in parity-check language. The central definitions are direct: Definition 1 and Eq. (3) set H_X=[I^T, 1+C], H_Z=[1+R, I] from an expanding matrix I with an order-l symmetry; the distance claims of Theorem 1 are then derived in Section II.C from the expansion inequalities, with explicit constants gamma_X and gamma_Z, and no parameter is fitted to the target distance. The asymptotic O([[N,N^{4/5},N^{3/5}]]) statement in Section IV.A combines Theorem 1 with the external LPS expander family [22] and an external distance-balancing construction [20]; no load-bearing step is justified solely by the authors' own prior work. The only self-citations are [26] (a SAT-solver distance verifier) and [27] (a repository of explicit examples); both are checkable, parameter-free tools and do not supply the theorem's premises, so they do not raise the circularity score. The explicit limitation in Section III—'But could there be anything shorter? We will not give the full details here'—means the claimed tilde_d_X = d_X d_C lower bound is asserted rather than proven in this review, but an unproved, externally cited gap is a completeness/correctness issue, not circularity: the balanced code's parameters are not defined in terms of its own conclusions, no fitted input is renamed as a prediction, and no uniqueness theorem is imported from the authors.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data; the constants alpha, beta, gamma_X, gamma_Z are existence and expanderness constants supplied by the graph family, not fit values. The main building blocks are external: LPS expanders, the CSS formalism, and good classical LDPC codes. No new physical entity or force is introduced.

assumptions (5)
  • standard math CSS stabilizer formalism: parity checks obey H_X H_Z^T = 0 (Eq. 2).
    Background used throughout to justify that the constructed operators commute and define a code.
  • domain assumption The matrix I admits an order-l permutation symmetry R,C such that R I = I C^T with all orbits of length l (Definition 1).
    This symmetry is what makes Eq. (3) satisfy the CSS condition; it is not derived, only illustrated on chosen graphs.
  • domain assumption I is (alpha,beta)-expanding with row and column weights at most s (Definition 2, Theorem 1).
    Expansion is the engine of the distance lower bounds; concrete families are imported from LPS expander graphs [22].
  • domain assumption A good classical LDPC code H_C with distance d_C = d_X/d_Z exists for distance balancing (Definition 3, Section III).
    The balancing step and its parameter tradeoffs come from [20]; the paper does not construct or prove this classical ingredient.
  • ad hoc to paper The orbit length l is odd when anti-commutation of the two logical operators is required (Section II.A).
    The paper states that provided l is odd, the two logical operators anti-commute; this parity condition is used to identify the logical qubits.

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Cite this review

Pith. "Pith review of Demystifying the Balanced Product Code: A Review." pith.science (2026). https://pith.science/paper/K7F4CBTW

@misc{pith2026250513679,
  author       = {Pith},
  title        = {Pith review of: Demystifying the Balanced Product Code: A Review},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K7F4CBTW}},
  note         = {Machine review of arXiv:2505.13679}
}
read the original abstract

The discovery of the family of balanced product codes was pivotal in the subsequent development of 'good' low density quantum error correcting codes that have optimal scaling of the key parameters of distance and storage density. We review this family, giving a completely different presentation to the original, minimising the abstraction and technicalities wherever possible. The target audience is anyone familiar with the stabilizer formalism for error correction wanting to understand how parity-check matrices can be constructed for high storage density quantum codes.

Figures

Figures reproduced from arXiv: 2505.13679 by the authors.

Figure 1
Figure 1. Depiction of code based on length 20 repetition [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Transversal Gates for Highly Asymmetric qLDPC Codes

    quant-ph 2025-06 conditional novelty 6.0 of 10

    First qLDPC code constructions with transversal non-Clifford phase gates, obtained by embedding a local code with the desired transversal gate into a Tanner-based hypergraph or balanced product code, at the cost of O(...

Reference graph

Works this paper leans on

38 extracted references · 21 canonical work pages · cited by 1 Pith paper

  1. [1]

    Hence, the number of edges is q(q2− 1)(p + 1)/2

    vertices, each of which has degree s = p + 1. Hence, the number of edges is q(q2− 1)(p + 1)/2. Being regu- lar graphs, we can take their incidence matrix I0. The graphs are expanding, and have an appropriate sym- metry of order l = q. The null space has dimension q(q2−1)(p−1)/2. We hold p fixed in order to ensure the low density property, and scaleq. When...

    work page

  2. [19]

    Bluvstein, S

    D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kali- nowski, D. Hangleiter, J. P. Bonilla Ataides, N. Maskara, I. Cong, X. Gao, P. Sales Rodriguez, T. Karolyshyn, G. Semeghini, M. J. Gullans, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Logical quantum processor based on reconfigurable atom arrays, Nature 626, ...

    2024

  3. [20]

    Tillich and G

    J.-P. Tillich and G. Zemor, Quantum LDPC codes with positive rate and minimum distance proportional to n 1 2 , IEEE Trans. Inform. Theory 60, 1193 (2014), arXiv:0903.0566 [quant-ph]

    arXiv 2014

  4. [22]

    Panteleev and G

    P. Panteleev and G. Kalachev, Quantum LDPC Codes with Almost Linear Minimum Distance, IEEE Trans. In- form. Theory 68, 213 (2022), arXiv:2012.04068 [quant- ph]

    arXiv 2022

  5. [2]

    In order to determine the edges of the graph, we define a set of matricesS3,5, with an edge between two vertices if their corresponding matrices u,v satisfy σu = v for some σ∈ S3,5

    For instance, one vertex, represented by 1, cor- responds to all the matrices {[ 1 0 0 1 ] , [ 2 0 0 2 ] , [ 3 0 0 3 ] , [ 4 0 0 4 ]} . In order to determine the edges of the graph, we define a set of matricesS3,5, with an edge between two vertices if their corresponding matrices u,v satisfy σu = v for some σ∈ S3,5. To that end, we consider the p + 1 = 4 ...

    work page

  6. [3]

    We have shown in the previous arguments that this is in no way necessary

    does not only use expander matrices, but also uses a local code, combining the two in a Tanner code [33]. We have shown in the previous arguments that this is in no way necessary. It does provide some advantage, however: if the local code has distance dl, then the X distance of the code is enhanced by a factor of approximately dl while the weights of some...

    work page

  7. [4]

    N. P. Breuckmann and J. N. Eberhardt, Balanced Prod- uct Quantum Codes, IEEE Transactions on Information Theory 67, 6653 (2021)

    2021

  8. [5]

    P. W. Shor, Polynomial-Time Algorithms for Prime Fac- torization and Discrete Logarithms on a Quantum Com- puter, SIAM J. Comput. 26, 1484 (1997)

    work page 1997

Show all 38 references
  1. [6]

    L. K. Grover, A Fast Quantum Mechanical Algorithm for Database Search, in Proceedings of the Twenty-eighth An- nual ACM Symposium on Theory of Computing , STOC ’96 (ACM, New York, NY, USA, 1996) pp. 212–219

    1996

  2. [7]

    A. W. Harrow, A. Hassidim, and S. Lloyd, Quantum Algorithm for Linear Systems of Equations, Phys. Rev. Lett. 103, 150502 (2009)

    2009

  3. [8]

    D. W. Berry, A. M. Childs, and R. Kothari, Hamilto- nian simulation with nearly optimal dependence on all parameters, in 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (2015) pp. 792–809, arXiv:1501.01715 [quant-ph]

    2015 arXiv

  4. [9]

    Arute and others, Quantum supremacy using a pro- grammable superconducting processor, Nature 574, 505 (2019)

    F. Arute and others, Quantum supremacy using a pro- grammable superconducting processor, Nature 574, 505 (2019)

    2019

  5. [10]

    Zhong and others, Quantum computational advan- tage using photons, Science 370, 1460 (2020)

    H.-S. Zhong and others, Quantum computational advan- tage using photons, Science 370, 1460 (2020)

    2020

  6. [11]

    L. S. Madsen and others, Quantum computational ad- vantage with a programmable photonic processor, Nature 606, 75 (2022)

    2022

  7. [12]

    V. V. Sivak, A. Eickbusch, B. Royer, S. Singh, I. Tsiout- sios, S. Ganjam, A. Miano, B. L. Brock, A. Z. Ding, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, and M. H. De- voret, Real-time quantum error correction beyond break- even, Nature 616, 50 (2023)

    2023

  8. [13]

    Laflamme, C

    R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, Per- fect Quantum Error Correcting Code, Phys. Rev. Lett. 77, 198 (1996)

    1996

  9. [14]

    A. M. Steane, Simple quantum error-correcting codes, Phys. Rev. A 54, 4741 (1996)

    1996

  10. [15]

    Aliferis, D

    P. Aliferis, D. Gottesman, and J. Preskill, Quantum accu- racy threshold for concatenated distance-3 codes, Quant. Inf. Comput. 6, 97 (2006)

    2006

  11. [16]

    Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303, 2 (2003)

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

    2003

  12. [17]

    Yamasaki and M

    H. Yamasaki and M. Koashi, Time-Efficient Constant- Space-Overhead Fault-Tolerant Quantum Computation, Nat. Phys. 20, 247 (2024)

    2024

  13. [18]

    Bravyi, D

    S. Bravyi, D. Poulin, and B. Terhal, Tradeoffs for reliable quantum information storage in 2D systems, Phys. Rev. Lett. 104, 050503 (2010), arXiv:0909.5200 [quant-ph]

    2010 arXiv

  14. [21]

    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 Sympo- sium on Theory of Computing , STOC 2022 (Association for Computing Machinery, New York, NY, USA, 2022) pp. 375–388, arXiv:21...

    2022 arXiv

  15. [23]

    S. Evra, T. Kaufman, and G. Z´ emor, Decodable Quan- tum LDPC Codes beyond the √n Distance Barrier Us- ing High-Dimensional Expanders, SIAM J. Comput. , FOCS20 (2022)

    2022

  16. [24]

    M. B. Hastings, J. Haah, and R. O’Donnell, Fiber Bun- dle Codes: Breaking the N 1/2 polylog(N) Barrier for Quantum LDPC Codes, in Proceedings of the 53rd An- nual ACM SIGACT Symposium on Theory of Computing (2021) pp. 1276–1288, arXiv:2009.03921 [quant-ph]

    2021 arXiv

  17. [25]

    Lubotzky, R

    A. Lubotzky, R. Phillips, and P. Sarnak, Ramanujan graphs, Combinatorica 8, 261 (1988)

    1988

  18. [26]

    Roffe, Quantum error correction: An introductory guide, Contemporary Physics 60, 226 (2019)

    J. Roffe, Quantum error correction: An introductory guide, Contemporary Physics 60, 226 (2019)

    2019

  19. [27]

    A. R. Calderbank and P. W. Shor, Good quantum error- correcting codes exist, Phys. Rev. A 54, 1098 (1996)

    1996

  20. [28]

    Beware: in what follows, the rows are not linearly inde- pendent

  21. [29]

    Kay and H

    A. Kay and H. Leitch, Sat-qecc, https://doi.org/10.17637/rh.29085299 (2025)

    2025 doi

  22. [30]

    Kay and H

    A. Kay and H. Leitch, Balanced Product Code Examples, https://doi.org/10.17637/rh.29085380 (2025)

    2025 doi

  23. [31]

    R. G. Gallager, Low-Density Parity-Check Codes (The MIT Press, 1963)

    1963

  24. [32]

    Indeed, this would still be true if we replaced 1 +C and 1 +R with f(C) and f(R) respectively, where f(x) is a low weight polynomial with binary coefficients

  25. [33]

    Assuming a large enough code 2 βγXl<αn that a token case can be neglected

  26. [34]

    This is often the only one

    Commonly, I has a fixed row weight, so it has a null vector comprising the all-ones. This is often the only one

  27. [35]

    Those vertices are identified by the 1 block

    From a programming perspective, apply row reduction modulo 2. Those vertices are identified by the 1 block

  28. [36]

    Tanner, A recursive approach to low complexity codes, IEEE Transactions on Information Theory27, 533 (1981)

    R. Tanner, A recursive approach to low complexity codes, IEEE Transactions on Information Theory27, 533 (1981)

    1981

  29. [37]

    For the sake of this example, we are not worrying about the two symmetry operators having different orders

  30. [38]

    shift in orbit

    H. Leitch and A. Kay, Transversal Gates for Highly Asymmetric qLDPC Codes, in preparation. 11 Appendix A: Where’s the Product? Throughout this paper, the product part of the balanced product code has been conspicuous in its absence. Where did it go? We have only considered the...

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