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arxiv: 2604.20838 · v2 · submitted 2026-04-22 · 🪐 quant-ph

Recognition: unknown

High-Girth Regular Quantum LDPC Codes from Affine-Coset Structures

Koki Okada , Kenta Kasai
Authors on Pith no claims yet

Pith reviewed 2026-05-10 00:18 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum LDPC codeshigh-girthaffine-cosetCPM liftCSS codesTanner graphbelief propagation
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The pith

A construction based on affine-coset structures produces high-girth regular quantum LDPC codes that lift to over 16,000 qubits with demonstrated decoding performance.

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

The paper establishes a method to create quantum low-density parity-check codes from a known base matrix pair that matches a product CSS code. An affine-coset description is used to prove that the associated Tanner graphs are exactly (3,8)-regular and have girth 8. Circulant permutation matrix lifting is then applied to generate larger codes, with detailed results for the case where the lift factor is 32. This produces a code with 16384 physical qubits, 4142 logical qubits, and distance at most 40, which a belief-propagation decoder with post-processing decodes to frame error rates near 10 to the minus 8 at a depolarizing error probability of 0.085.

Core claim

We construct a quantum low-density parity-check code family from a length-512 Calderbank-Shor-Steane base matrix pair. The base pair is permutation-equivalent to the known SPC(3) product CSS code, and the present affine-coset description gives a direct proof that both Tanner graphs are (3,8)-regular with girth 8. The base code has parameters [[512,174,8]]. We then apply circulant permutation matrix (CPM) lifts. The main decoding experiment uses the CPM-lifted code with lift factor P=32, which has parameters [[16384,4142,≤40]], under the code-capacity depolarizing model. A belief-propagation decoder with post-processing achieved frame error rate about 10^{-8} at p=0.085, and one observed лог

What carries the argument

The affine-coset description of the base matrix pair that proves (3,8)-regularity and girth 8, combined with circulant permutation matrix lifting to scale the code size.

If this is right

  • The construction produces a family of (3,8)-regular quantum LDPC codes with girth 8 at different lift factors.
  • The lifted code with P=32 has parameters [[16384,4142,≤40]].
  • Belief-propagation decoding with post-processing achieves frame error rate about 10^{-8} at depolarizing probability p=0.085.
  • The distance of the lifted code is at most 40 based on the observed logical error of weight 40.

Where Pith is reading between the lines

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

  • If the girth property is preserved under lifting, these codes may support reliable error correction at larger scales.
  • The upper bound on distance from decoder observations could be tightened by searching for minimum-weight logical operators.
  • Similar affine-coset constructions might apply to other base codes to generate additional high-girth quantum LDPC families.
  • Performance at higher noise levels or with different decoders remains to be explored for these codes.

Load-bearing premise

The length-512 base matrix pair is permutation-equivalent to the SPC(3) product CSS code so that the affine-coset description can rigorously establish the regularity and girth.

What would settle it

Computing all cycles in the Tanner graph of the base code and finding any of length 4 or 6, or identifying a logical operator of weight less than 40 in the P=32 lifted code.

Figures

Figures reproduced from arXiv: 2604.20838 by Kenta Kasai, Koki Okada.

Figure 1
Figure 1. Figure 1: Let e1, . . . , e9 be the standard basis of F 9 2 , and set (a1, a2, a3, b1, b2, b3, c1, c2, c3) = (e1, e2, e3, e4, e5, e6, e7, e8, e9). Then A = ⟨e1, e2, e3⟩, B = ⟨e4, e5, e6⟩, C = ⟨e7, e8, e9⟩, and D1 = ⟨e1, e4, e7⟩, D2 = ⟨e2, e5, e8⟩, D3 = ⟨e3, e6, e9⟩. The columns are ordered as follows. A qubit is a vector x = X 3 i=1 αiai + X 3 i=1 βibi + X 3 i=1 γici ∈ V, and we interpret (α1, α2, α3, β1, β2, β3, γ1… view at source ↗
Figure 1
Figure 1. Figure 1: The other matrices M(B), M(C), M(D1), M(D2), M(D3) are formed by the same incidence rule. Stacking them as Hbase X =   M(A) M(B) M(C)   , Hbase Z =   M(D1) M(D2) M(D3)   gives the base matrix pair displayed in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Visualization of the nonzero entries of the base matrices [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FER for the P = 32 CPM-lifted code [[16384, 4142, ≤ 40]] under the code-capacity depolarizing model. The curve shows the BP decoder with post-processing, and the shaded band gives the 95% Wilson confidence interval. The red dashed line is the hashing bound for rate 1/4, and the black dash-dotted line is a (3, 8)-regular DE reference for a random sparse-graph ensemble, not a threshold for a CSS-orthogonal e… view at source ↗
read the original abstract

We construct a quantum low-density parity-check code family from a length-$512$ Calderbank--Shor--Steane base matrix pair. The base pair is permutation-equivalent to the known SPC(3) product CSS code, and the present affine-coset description gives a direct proof that both Tanner graphs are $(3,8)$-regular with girth $8$. The base code has parameters $[[512,174,8]]$. We then apply circulant permutation matrix (CPM) lifts. The main decoding experiment uses the CPM-lifted code with lift factor $P=32$, which has parameters $[[16384,4142,\le 40]]$, under the code-capacity depolarizing model. A belief-propagation decoder with post-processing achieved frame error rate about $10^{-8}$ at $p=0.085$, and one observed logical residual of weight $40$ gives a decoder-derived upper bound $d\le 40$.

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

1 major / 2 minor

Summary. The manuscript constructs a family of quantum LDPC codes from a length-512 CSS base matrix pair that is permutation-equivalent to the known SPC(3) product code. An affine-coset description is used to prove that both Tanner graphs are (3,8)-regular with girth exactly 8, yielding base parameters [[512,174,8]]. CPM lifts are then applied; the main result is the P=32 lifted code with parameters [[16384,4142,≤40]]. Under the code-capacity depolarizing model, belief-propagation decoding with post-processing achieves a frame error rate of approximately 10^{-8} at p=0.085, and a single observed logical residual of weight 40 supplies the decoder-derived upper bound d≤40.

Significance. If the construction and reported performance hold, the work supplies an explicit, deterministic method for generating regular high-girth QLDPC codes via affine-coset structures and standard CPM lifting. The direct girth and regularity proof, the absence of free parameters in the construction, and the competitive empirical decoding result at block length 16384 are strengths that would advance the design of practical quantum LDPC codes.

major comments (1)
  1. [Abstract and decoding experiments] The frame-error-rate claim of ~10^{-8} at p=0.085 and the distance upper bound d≤40 both rest on a single decoding run and one verified logical residual. No number of Monte-Carlo trials, confidence intervals, or additional independent residuals are reported, which limits the statistical support for the performance statements in the abstract and decoding section.
minor comments (2)
  1. [Base-code construction] The permutation-equivalence of the length-512 base matrix to the SPC(3) product CSS code is asserted but would benefit from an explicit reference or short proof sketch in the base-code section to allow immediate verification of the [[512,174,8]] parameters.
  2. [Lifting construction] Notation for the lift factor (denoted P=32) should be introduced once and used consistently when stating the lifted parameters [[16384,4142,≤40]].

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, the positive overall assessment, and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and decoding experiments] The frame-error-rate claim of ~10^{-8} at p=0.085 and the distance upper bound d≤40 both rest on a single decoding run and one verified logical residual. No number of Monte-Carlo trials, confidence intervals, or additional independent residuals are reported, which limits the statistical support for the performance statements in the abstract and decoding section.

    Authors: We agree that the current text provides insufficient detail on the Monte-Carlo setup. In the revised manuscript we will explicitly state the total number of trials performed, report the resulting empirical frame-error-rate with the observed count of errors, and add a brief discussion of the statistical uncertainty (e.g., via the rule-of-three or Poisson confidence interval for a single observed event). For the distance claim we will clarify that d ≤ 40 is a decoder-derived upper bound obtained from the weight of the single observed logical residual; we will note that exhaustive minimum-distance computation is intractable at this block length and that such decoder-based bounds are standard in the literature. These additions will be confined to the decoding-experiments section and the abstract, preserving the original claims while improving transparency. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from known base code and standard lifting

full rationale

The paper explicitly starts from a known SPC(3) product CSS code (permutation-equivalent base pair with parameters [[512,174,8]]), supplies an independent affine-coset description that directly establishes (3,8)-regularity and girth 8 without any fitted parameters or self-referential definitions, then applies standard CPM lifting whose dimension and distance bounds follow from arithmetic and one observed logical operator. Empirical decoder performance (FER ~10^{-8} at p=0.085) is simulation output, not a constructed prediction. No load-bearing self-citations, ansatzes, or reductions of claims to their own inputs appear; the chain remains externally grounded.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the equivalence of the base pair to a known product code and on standard properties of CPM lifts; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The given length-512 base matrix pair is permutation-equivalent to the SPC(3) product CSS code
    Stated directly in the abstract as the starting point for the construction.
  • domain assumption CPM lifts of the base preserve regularity and produce the claimed parameters and girth properties
    Invoked when moving from the base [[512,174,8]] code to the lifted [[16384,4142,≤40]] code.

pith-pipeline@v0.9.0 · 5461 in / 1306 out tokens · 42220 ms · 2026-05-10T00:18:23.024285+00:00 · methodology

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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. High-Girth Regular Quantum LDPC Codes from Square-Base Hypergraph Products via CPM Lifts

    quant-ph 2026-04 conditional novelty 6.0

    Constructs explicit regular high-girth quantum LDPC codes from square-base hypergraph products and CPM lifts, including a [[28800,62]] (3,6)-regular code with zero observed decoding failures in 2.993e8 trials at p=0.1402.

Reference graph

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29 extracted references · 5 canonical work pages · cited by 1 Pith paper · 2 internal anchors

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