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arxiv: 2606.17268 · v1 · pith:QJKXJHCWnew · submitted 2026-06-15 · 🪐 quant-ph · cs.IT· math.IT

Breaking the bicycle frame: Coset-based quantum LDPC codes

Arda Aydin , Itzhak Tamo , Alexander Barg This is my paper

Pith reviewed 2026-06-27 03:07 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT
keywords quantum LDPC codescoset actionstwo-block constructionsstabilizer codessyndrome extractionBP-OSD decodinggroup actionsquantum error correction
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The pith

Group actions on cosets of subgroups produce quantum LDPC codes outside the two-block group algebra family.

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

The paper generalizes two-block group algebra codes by replacing regular group actions with the action of a group on the cosets of one of its subgroups. This change enlarges the space of admissible constructions and lets a computer search locate new quantum LDPC codes with previously unreported parameters. Explicit examples include weight-6 codes [[48,8,6]], [[96,8,10]], [[224,12,16]] and weight-8 codes [[84,16,8]], [[112,16,10]], [[128,16,12]], [[168,16,15]]. The same family admits a syndrome extraction circuit of depth w+2 and, when decoded with BP-OSD, reaches thresholds near 0.65 percent for the weight-6 members and 0.35 percent for the weight-8 members under circuit-level noise.

Core claim

Generalizing the construction of two-block group algebra (2BGA) codes, we introduce a family of two-block quantum LDPC codes constructed using the action of a group on the cosets of its subgroup. This replaces the regular group actions of the earlier two-block constructions and significantly expands the search space, yielding new quantum LDPC codes outside the 2BGA family. Through a computer search, we identify several new quantum LDPC codes with the listed parameters. We introduce a maximally packed syndrome extraction schedule of depth w+2 for any code with maximum stabilizer weight w from our family and show competitive performance with BB codes under circuit-level noise when decoded with

What carries the argument

The action of a group G on the left cosets of a subgroup H, used to define the two blocks of the quantum stabilizer code.

If this is right

  • New quantum LDPC codes exist with parameters [[48,8,6]], [[96,8,10]], [[224,12,16]], [[84,16,8]], [[112,16,10]], [[128,16,12]], and [[168,16,15]].
  • A syndrome extraction schedule whose depth is only two more than the maximum stabilizer weight is valid for every code in the family.
  • The weight-6 subfamily reaches an error threshold of approximately 0.65 percent and the weight-8 subfamily reaches approximately 0.35 percent under circuit-level noise with BP-OSD decoding.
  • A group-theoretic construction generates infinite sequences of graph-based covers of 2BGA codes and recovers earlier results on such covers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Varying the ambient group and the subgroup may systematically produce longer or higher-distance members of the same family.
  • The coset formalism could be applied to other algebraic constructions of quantum or classical LDPC codes that rely on group actions.
  • The reported thresholds indicate that these codes remain candidates for near-term hardware if the block length can be increased while preserving the relative distance.

Load-bearing premise

The chosen coset actions must produce sets of commuting Pauli operators whose common +1 eigenspace has exactly the distance stated for each listed code.

What would settle it

A direct computation showing that the minimum distance of the [[224,12,16]] code is smaller than 16, or a circuit-level simulation in which the BP-OSD threshold for the weight-8 family falls below 0.3 percent.

Figures

Figures reproduced from arXiv: 2606.17268 by Alexander Barg, Arda Aydin, Itzhak Tamo.

Figure 1
Figure 1. Figure 1: FIG. 1: Various quantum LDPC codes and their relationship with each other. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Logical error rate [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Logical error rate [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
read the original abstract

Generalizing the construction of two-block group algebra (2BGA) codes, we introduce a family of two-block quantum LDPC codes constructed using the action of a group on the cosets of its subgroup. This replaces the regular group actions of the earlier two-block constructions and significantly expands the search space, yielding new quantum LDPC codes outside the 2BGA family. Through a computer search, we identify several new quantum LDPC codes, including weight-6 codes with parameters $[[48,8,6]]$, $[[96,8,10]]$, and $[[224,12,16]]$, as well as weight-8 codes with parameters $[[84,16,8]]$, $[[112,16,10]]$, $[[128,16,12]]$, and $[[168,16,15]]$. Furthermore, we introduce a maximally packed syndrome extraction schedule of depth $w+2$, including initialization and measurement steps, for any code with a maximum stabilizer weight of $w$ from our family. Under a standard circuit-level noise model, our codes, when decoded using BP-OSD, perform competitively with BB codes, achieving thresholds of $\approx0.65\%$ for the weight-6 family and $\approx0.35\%$ for the weight-8 family. Finally, we introduce a group-theoretic framework to generate sequences of graph-based covers of 2BGA codes, recovering and extending recent results on code constructions of this type.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper generalizes two-block group algebra (2BGA) quantum LDPC codes to a coset-based construction in which a group acts on the cosets of a subgroup, thereby enlarging the search space. A computer search yields new codes outside the 2BGA family, including weight-6 examples with parameters [[48,8,6]], [[96,8,10]], [[224,12,16]] and weight-8 examples with parameters [[84,16,8]], [[112,16,10]], [[128,16,12]], [[168,16,15]]. The work also presents a maximally packed syndrome-extraction circuit of depth w+2, reports BP-OSD thresholds of approximately 0.65% (weight-6) and 0.35% (weight-8) under circuit-level noise, and supplies a group-theoretic framework that generates sequences of graph-based covers of 2BGA codes.

Significance. If the reported parameters are correct and the codes lie outside prior families, the coset construction meaningfully enlarges the known landscape of quantum LDPC codes and supplies concrete examples with competitive distances and thresholds. The syndrome schedule and cover framework are practical contributions that could be adopted independently of the specific codes found.

major comments (2)
  1. [Computer-search section (results reported in abstract and §4–5)] Computer-search section (results reported in abstract and §4–5): the central claim that the listed parameters are achieved by valid CSS codes outside the 2BGA family rests on unverified computational output. Explicit group/subgroup pairs, the resulting parity-check matrices, or a certified minimum-weight enumeration procedure must be supplied so that H_X H_Z^T = 0 and the stated distances can be independently recomputed; without them the numerical results cannot be treated as load-bearing evidence.
  2. [§6 (performance evaluation)] §6 (performance evaluation): the reported thresholds are obtained with BP-OSD on the new codes, yet no comparison is made against the same decoder applied to the nearest 2BGA codes of comparable length and rate; this leaves open whether the observed competitiveness is due to the coset construction or to decoder tuning.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'weight-6 codes' is ambiguous; it should explicitly state that the maximum stabilizer weight is 6.
  2. [Construction section] Notation for the coset action is introduced without a compact tabular summary of the group orders and indices used in the search; adding such a table would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and helpful comments. Below we provide point-by-point responses to the major comments, and we will make the suggested revisions to the manuscript.

read point-by-point responses

  1. Referee: Computer-search section (results reported in abstract and §4–5): the central claim that the listed parameters are achieved by valid CSS codes outside the 2BGA family rests on unverified computational output. Explicit group/subgroup pairs, the resulting parity-check matrices, or a certified minimum-weight enumeration procedure must be supplied so that H_X H_Z^T = 0 and the stated distances can be independently recomputed; without them the numerical results cannot be treated as load-bearing evidence.

    Authors: We fully agree with this assessment. The computer search results form a key part of our contribution, and independent verification is necessary. In the revised version, we will provide the explicit group/subgroup pairs for each reported code, the corresponding parity-check matrices, and details on the minimum-weight enumeration method used. This will enable readers to confirm the CSS property and the distances. revision: yes

  2. Referee: §6 (performance evaluation): the reported thresholds are obtained with BP-OSD on the new codes, yet no comparison is made against the same decoder applied to the nearest 2BGA codes of comparable length and rate; this leaves open whether the observed competitiveness is due to the coset construction or to decoder tuning.

    Authors: We appreciate this point. To address it, we will add a comparison in the revised manuscript by applying the BP-OSD decoder to 2BGA codes with similar lengths and rates under the same circuit-level noise model. This will help isolate the benefits of the coset construction. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit coset construction and search are independent of inputs

full rationale

The paper defines a new family of two-block quantum LDPC codes via the action of a group on cosets of a subgroup, explicitly generalizing (but not presupposing) prior 2BGA constructions. Parameters such as [[48,8,6]] are obtained by computer enumeration over this expanded space, with no equations or claims that reduce a derived quantity to a fitted parameter or self-citation by construction. The syndrome schedule and performance results are likewise downstream of the explicit matrices produced by the search. No load-bearing step matches any of the six enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities beyond the coset construction itself.

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Forward citations

Cited by 1 Pith paper

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

  1. Topological Codes from Space Groups: A Route beyond Translation Invariance

    quant-ph 2026-06 unverdicted novelty 6.0

    Framework constructs CSS topological codes from space groups via invariant theory, giving criteria for topological order and anyon counting while claiming enhanced locality.

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    Proof of Proposition II.1 First, let us prove that𝐿(𝑔) permutes the cosets in 𝐺/𝐻for all𝑔∈𝐺. Let𝑔∈𝐺. The following chain of relationships 𝑥𝐻=𝑦𝐻⇐ ⇒𝑥 −1𝑦∈𝐻 ⇐ ⇒𝑥 −1𝑦=𝑥 −1𝑔−1𝑔𝑦= (𝑔𝑥) −1(𝑔𝑦)∈𝐻 ⇐ ⇒(𝑔𝑥)𝐻= (𝑔𝑦)𝐻 ⇐ ⇒𝐿(𝑔)(𝑥𝐻) =𝐿(𝑔)(𝑦𝐻), shows that𝐿(𝑔) is well defined by reading it from left to right, and it is an injection by reading it from right to left. Hence,𝐿(...

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    Then 𝐿(𝑔1)∘𝑅(𝑔 2)(𝑥𝐻) =𝐿(𝑔 1)((𝑥𝑔2)𝐻) = (𝑔1(𝑥𝑔2))𝐻 = ((𝑔1𝑥)𝑔2)𝐻 =𝑅(𝑔 2)((𝑔1𝑥)𝐻) =𝑅(𝑔 2)∘𝐿(𝑔 1)(𝑥𝐻)

    Proof of Proposition II.2 Let𝑥𝐻be any left coset of𝐻in𝐺. Then 𝐿(𝑔1)∘𝑅(𝑔 2)(𝑥𝐻) =𝐿(𝑔 1)((𝑥𝑔2)𝐻) = (𝑔1(𝑥𝑔2))𝐻 = ((𝑔1𝑥)𝑔2)𝐻 =𝑅(𝑔 2)((𝑔1𝑥)𝐻) =𝑅(𝑔 2)∘𝐿(𝑔 1)(𝑥𝐻)

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    We have, for all𝑥∈𝐺, 𝑔∈ker𝐿⇐ ⇒𝐿(𝑔)(𝑥𝐻) =𝑥𝐻 ⇐ ⇒(𝑔𝑥)𝐻=𝑥𝐻 ⇐ ⇒𝑥 −1(𝑔𝑥)∈𝐻 ⇐ ⇒𝑔∈𝑥𝐻𝑥 −1 ⇐ ⇒𝑔∈ ⋂︁ 𝑥∈𝐺 𝑥𝐻𝑥 −1 ⇐ ⇒𝑔∈Core 𝐺(𝐻)

    Proof of Proposition II.3 [23, Thm.4.2.3] Once we show that ker𝐿= Core 𝐺(𝐻), the claim Im𝐿 ∼= 𝐺/ker𝐿will follow by the first isomorphism the- orem. We have, for all𝑥∈𝐺, 𝑔∈ker𝐿⇐ ⇒𝐿(𝑔)(𝑥𝐻) =𝑥𝐻 ⇐ ⇒(𝑔𝑥)𝐻=𝑥𝐻 ⇐ ⇒𝑥 −1(𝑔𝑥)∈𝐻 ⇐ ⇒𝑔∈𝑥𝐻𝑥 −1 ⇐ ⇒𝑔∈ ⋂︁ 𝑥∈𝐺 𝑥𝐻𝑥 −1 ⇐ ⇒𝑔∈Core 𝐺(𝐻). Similarly, ker𝑅=𝐻since 𝑔∈ker𝑅⇐ ⇒𝑅(𝑔)(𝑥𝐻) =𝑥𝐻 ⇐ ⇒(𝑥𝑔)𝐻=𝑥𝐻 ⇐ ⇒𝑥 −1(𝑥𝑔)∈𝐻 ⇐ ⇒𝑔∈𝐻. Appendix B: D...