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arxiv: 2606.27130 · v1 · pith:OFUEUKU3new · submitted 2026-06-25 · 🪐 quant-ph · cs.IT· math.IT

Rate-2/3 Girth-8 (3,18)-Regular Quantum LDPC Codes from Two-Branch Finite-Field Bases and CPM Lifts

Koki Okada , Kenta Kasai This is my paper

Pith reviewed 2026-06-26 04:36 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT
keywords quantum LDPC codesCSS codesfinite-field basesCPM liftsgirth-8 Tanner graphsbelief propagation decodingrate 2/3error correction performance
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The pith

A (3,18)-regular two-branch finite-field base with CPM lift of degree 101 produces a rate-2/3 CSS quantum LDPC code with parameters [[34542,23032,d≤310]].

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

The paper constructs a quantum LDPC code by starting from a (3,18)-regular two-branch finite-field base matrix and applying a circulant-permutation-matrix lift of size 101. This produces the parity-check matrices H_X and H_Z of a CSS code whose Tanner graphs each have girth 8. The resulting code has block length 34542, dimension 23032, and an upper bound on minimum distance of 310. Joint log-likelihood-ratio belief propagation decoding with deterministic post-processing shows no failures across 10^8 trials at error probability 0.01, with the frame error rate transition estimated near 0.029.

Core claim

We construct a rate-2/3 quantum low-density parity-check (LDPC) code from a (3,18)-regular two-branch finite-field base and a circulant-permutation-matrix (CPM) lift of degree P=101. The resulting code is a Calderbank-Shor-Steane (CSS) code with parameters [[34542,23032,d≤310]]. The construction has row weight 18 and column weight 3, and the Tanner graphs of H_X and H_Z separately have girth 8. Decoder experiments with log-likelihood-ratio (LLR) joint belief propagation (BP) and deterministic post-processing show no failures in 10^8 trials at p=0.01, and a finite-length frame error rate (FER) sweep estimates the transition near p=0.029.

What carries the argument

Two-branch finite-field base lifted by circulant-permutation-matrix of degree 101, which produces separate girth-8 Tanner graphs for H_X and H_Z while satisfying the CSS orthogonality condition.

Load-bearing premise

The two-branch finite-field base is (3,18)-regular and the CPM lift of degree 101 produces separate girth-8 Tanner graphs for H_X and H_Z that satisfy the CSS condition.

What would settle it

A single decoding failure in 10^8 trials at p=0.01 or identification of a logical operator of weight below the reported upper bound of 310 would contradict the stated performance and distance claims.

Figures

Figures reproduced from arXiv: 2606.27130 by Kenta Kasai, Koki Okada.

Figure 1
Figure 1. Figure 1: Block-colored bitmap of the base parity-check matrices [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FER measurements for the constructed (3, 18), P = 101 lifted code. The horizontal axis is shown on 0 ≤ p ≤ 0.1. The vertical lines mark the BP threshold estimate p = 0.034 and the rate-2/3 hashing bound p = 0.04454. The markers are connected as a visual guide. The ETS-library branch was tested on failure instances obtained in the p = 0.0237 measure￾ments. Five of these failures had Z-side residual-syndrome… view at source ↗
read the original abstract

We construct a rate-$2/3$ quantum low-density parity-check (LDPC) code from a $(3,18)$-regular two-branch finite-field base and a circulant-permutation-matrix (CPM) lift of degree $P=101$. The resulting code is a Calderbank-Shor-Steane (CSS) code with parameters $[[34542,23032,d\le 310]]$. We do not regard this upper bound as an estimate of the true minimum distance; rather, $d\le310$ is the tightest upper bound currently obtained from structural lifts and decoder-produced logical errors. The construction has row weight 18 and column weight 3, and the Tanner graphs of $H_X$ and $H_Z$ separately have girth 8. Decoder experiments with log-likelihood-ratio (LLR) joint belief propagation (BP) and deterministic post-processing show no failures in $10^8$ trials at $p=0.01$, and a finite-length frame error rate (FER) sweep estimates the transition near $p=0.029$.

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 manuscript constructs a rate-2/3 girth-8 (3,18)-regular quantum LDPC code from a two-branch finite-field base and a CPM lift of degree P=101, yielding the CSS code [[34542,23032,d≤310]] with separate girth-8 Tanner graphs for H_X and H_Z. Decoder trials with LLR joint BP plus post-processing report zero failures in 10^8 trials at p=0.01 and an FER transition near p=0.029; d≤310 is explicitly an upper bound from structural lifts and observed logical errors, not a distance estimate.

Significance. If the algebraic construction and girth/CSS properties hold, the work supplies one of the largest explicit high-rate quantum LDPC codes with column weight 3 and girth 8, together with concrete Monte-Carlo evidence of strong finite-length performance. Explicit constructions of this scale are useful benchmarks and potential building blocks for quantum error correction.

major comments (2)
  1. [Construction] Construction paragraph / base definition: the two-branch finite-field base must be stated explicitly (field, branch matrices, or generator) so that the claimed (3,18)-regularity, the CSS condition H_X H_Z^T=0 after the P=101 CPM lift, and the exact block length 34542 can be independently verified.
  2. [Results] Results / distance paragraph: the upper bound d≤310 is obtained from structural lifts and decoder-produced logical errors; the precise logical operators or lift argument that establishes this bound (and why it is tightest) must be given to support the stated claim.
minor comments (2)
  1. [Abstract] Abstract: the FER transition estimate near p=0.029 should be accompanied by the number of Monte-Carlo trials and any error bars or confidence intervals used.
  2. [Decoder experiments] Decoder section: the joint BP + deterministic post-processing procedure should be described with sufficient pseudocode or parameter settings (e.g., iteration limit, post-processing rule) to allow reproduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation for minor revision. We address each major comment below.

read point-by-point responses

  1. Referee: Construction paragraph / base definition: the two-branch finite-field base must be stated explicitly (field, branch matrices, or generator) so that the claimed (3,18)-regularity, the CSS condition H_X H_Z^T=0 after the P=101 CPM lift, and the exact block length 34542 can be independently verified.

    Authors: We agree that an explicit statement of the two-branch finite-field base will improve independent verification. The revised manuscript will add the specific finite field, the two branch matrices (or generator), and the precise parameters used to construct the base. This addition will directly confirm the (3,18)-regularity, the CSS orthogonality after the degree-101 CPM lift, and the resulting block length of 34542. revision: yes

  2. Referee: Results / distance paragraph: the upper bound d≤310 is obtained from structural lifts and decoder-produced logical errors; the precise logical operators or lift argument that establishes this bound (and why it is tightest) must be given to support the stated claim.

    Authors: We will expand the distance paragraph in the revision to include a concise description of the structural-lift argument and the specific decoder-observed logical errors that yield the upper bound d≤310, together with a short explanation of why this is the tightest bound obtained so far. The manuscript already states that d≤310 is an upper bound rather than a distance estimate; the added text will make the supporting argument explicit without altering that characterization. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit algebraic construction

full rationale

The paper describes a direct constructive procedure: select a (3,18)-regular two-branch finite-field base matrix and apply a CPM lift of degree 101 to obtain explicit parity-check matrices H_X and H_Z satisfying the CSS orthogonality condition, girth-8 property, and the stated length/rate. No equations redefine outputs in terms of fitted inputs, no predictions are made from subsets of the same data, and no load-bearing uniqueness or ansatz is imported via self-citation. The reported parameters and decoder performance follow immediately from the algebraic definition of the lift; the derivation chain is self-contained and does not reduce to its own inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Based on abstract only; the construction rests on algebraic properties of finite fields and the lifting procedure. Full details unavailable.

free parameters (1)
  • lift degree P = 101
    Chosen parameter for the CPM lift that determines the final block length and rate.
axioms (2)
  • domain assumption Finite fields admit two-branch bases yielding (3,18)-regular bipartite graphs suitable for quantum CSS codes
    Invoked to define the base matrix before lifting.
  • domain assumption CPM lifts of the base preserve column/row weights and produce girth-8 Tanner graphs for both H_X and H_Z
    Required to claim the girth-8 property and CSS validity.

pith-pipeline@v0.9.1-grok · 5735 in / 1533 out tokens · 60037 ms · 2026-06-26T04:36:55.943059+00:00 · methodology

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Reference graph

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