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arxiv: 2502.17052 · v4 · submitted 2025-02-24 · 🪐 quant-ph

Existence and Characterisation of Bivariate Bicycle Codes

Jasper Johannes Postema , Servaas J.J.M.F. Kokkelmans This is my paper

Pith reviewed 2026-05-23 02:44 UTC · model grok-4.3

classification 🪐 quant-ph
keywords bivariate bicycle codesquantum error correctionLDPC codesring structureasymptotic badnesscode dimensionexistence conditionsquantum memory
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The pith

Bivariate bicycle codes have dimensions and existence conditions predicted from ring structure but are asymptotically bad.

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

The paper establishes that the ring structure of bivariate bicycle codes permits closed-form prediction of their dimension together with explicit conditions on when they exist. A sympathetic reader would care because these parameters determine whether the codes can deliver compact quantum memory with low overhead and error-correction performance beyond current topological codes. The same ring analysis shows the codes are asymptotically bad: both rate and relative distance vanish as block length grows. This rules the family out of the search for asymptotically good LDPC quantum codes while leaving known moderate-length instances available for experiments.

Core claim

Leveraging their ring structure, the dimension of bivariate bicycle codes can be predicted in closed form and conditions for their existence derived. The codes are asymptotically bad, so this subclass cannot contribute to the search for practical good low-density parity-check codes with non-vanishing rate and relative distance, although known moderate-length examples remain usable for experimental demonstrations of improved quantum error correction.

What carries the argument

The ring structure of bivariate bicycle codes, which supplies closed-form expressions for dimension and existence conditions.

If this is right

  • Dimension is obtained directly from the ring without constructing the full parity-check matrix.
  • Existence is governed by explicit algebraic conditions derived from the same ring.
  • Asymptotic badness excludes the family from the search for good LDPC quantum codes.
  • Moderate-length instances retain experimental utility for demonstrating QEC beyond the surface code.

Where Pith is reading between the lines

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

  • The ring-based method may apply to other quasi-cyclic or polynomial-based quantum code families.
  • Alternative constructions outside the bivariate polynomial ring will be required to reach asymptotically good parameters.
  • Direct verification of the dimension formula on the smallest published BB codes would provide an immediate consistency check.

Load-bearing premise

The ring structure of bivariate bicycle codes permits closed-form prediction of dimension and existence without additional fitting or post-hoc selection.

What would settle it

A concrete bivariate bicycle code whose dimension computed from the parity-check matrix differs from the ring-structure formula, or an infinite family with non-vanishing rate and distance linear in block length.

Figures

Figures reproduced from arXiv: 2502.17052 by Jasper Johannes Postema, Servaas J.J.M.F. Kokkelmans.

Figure 1
Figure 1. Figure 1: Tanner graph of a BB code, where black squares denote data qubits, and blue/golden squares denote stabiliser qubits. Two polyno￾mials a, b define the overall connectivity. The trinomial ansatz enforces the code to be LDPC with weight-6 stabilisers. There always exists a graph permutation such that one can rearrange the grid in such a way that every qubit has four nearest neighbours in the 2D plane, and two… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Histograms of coprime BB codes, where the relative minimum distance d/n is graphed against the code length n. Known codes (see Ref. [14]) are coloured green, and our new codes are high￾lighted in blue. The asymptotic scaling (dashed line) is plotted to show the behaviour upper bound, and serves as a highlight of the asymptotic behaviour rather than a strict inequality. For larger code lengths, it appea… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of equivalent codes between the [[30, 4, 4]] and [[30, 4, 6]] BB codes (highlighted in solid lines), and 4 patches of the [[9, 1, 3]], [[25, 1, 5]] and [[49, 1, 7]] surface codes (highlighted in dashed lines). Codes with equivalent slopes will yield roughly the same perfor￾mance per logical qubit, though BB codes trade physical qubit overhead for harder connectivity constraints. As an example, t… view at source ↗
read the original abstract

Encoding quantum information in a quantum error correction (QEC) code offers protection against decoherence and enhances the fidelity of qubits and gate operations. One of the fundamental challenges of QEC is to construct codes with asymptotically good parameters, i.e. a non-vanishing rate and relative minimum distance. Such codes provide compact quantum memory with low overhead and enhanced error correcting capabilities, compared to state-of-the-art topological error correction codes such as the surface or colour codes. Recently, bivariate bicycle (BB) codes have emerged as a promising candidate for such compact memory, though the exact tradeoff of the code parameters $[[n,k,d]]$ remained unknown. In this Article, we explore these codes by leveraging their ring structure, and predict their dimension as well as conditions on their existence. Finally, we highlight asymptotic badness. Though this excludes this subclass of codes from the search towards practical good low-density parity check (LDPC) codes, it does not affect the utility of the moderately long codes that are known, which can already be used to experimentally demonstrate better QEC beyond the surface code.

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 claims that bivariate bicycle codes, defined via multiplication operators in the bivariate polynomial ring over F_2, admit an algebraic characterization that yields explicit predictions for the code dimension k together with necessary and sufficient conditions for existence; it further shows that the resulting family is asymptotically bad (vanishing rate or relative distance) and therefore cannot furnish asymptotically good quantum LDPC codes, while still endorsing the practical utility of known moderate-length instances.

Significance. If the ring-theoretic predictions are both correct and closed-form, the work supplies a concrete algebraic handle on a recently proposed family of compact quantum codes, directly addressing the open question of their [[n,k,d]] trade-off and supplying a negative result on asymptotic goodness that narrows the search space for good quantum LDPC codes.

major comments (2)
  1. [Ring-structure leverage paragraph / dimension-prediction derivation] The central derivation of the dimension formula (the paragraph immediately following the ring-structure definition) asserts a closed-form expression obtained directly from the module corank. However, the corank of the map defined by the pair of bivariate polynomials is the degree of their gcd in the ring F_2[x,y]/(x^a-1,y^b-1); the Euclidean algorithm that computes this gcd produces an output whose length and explicit form depend on the factorization of the input polynomials. No uniform, input-independent closed-form expression is exhibited, nor is it shown that the existence conditions avoid post-hoc enumeration of factor pairs. This directly affects the load-bearing claim that the ring structure alone supplies the prediction without additional fitting or case-by-case selection.
  2. [Asymptotic-badness section] The asymptotic-badness argument (final section) concludes that the family cannot be asymptotically good. The argument appears to rest on the dimension formula derived above; if that formula is only computable per instance rather than closed-form, the uniform statement that the rate vanishes for all sequences of (a,b) with n=2ab growing requires an additional uniform bound on the gcd degree that is not supplied.
minor comments (2)
  1. [Definition of the codes] Notation for the two generating polynomials is introduced without an explicit equation label; subsequent references would be clearer if each polynomial were assigned a numbered display equation.
  2. [Conclusion paragraph] The statement that known moderate-length codes 'can already be used to experimentally demonstrate better QEC' would benefit from a single citation to the relevant experimental work or a brief parenthetical remark on the relevant code lengths.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. We address the two major comments point by point below, clarifying the algebraic nature of our results while acknowledging where additional exposition may strengthen the manuscript.

read point-by-point responses

  1. Referee: The central derivation of the dimension formula asserts a closed-form expression obtained directly from the module corank. However, the corank is the degree of their gcd in the ring; the Euclidean algorithm produces an output whose length depends on the factorization. No uniform, input-independent closed-form expression is exhibited, nor is it shown that the existence conditions avoid post-hoc enumeration of factor pairs.

    Authors: The dimension is given explicitly by the uniform formula k = 2 deg(gcd(f,g)) in the quotient ring F_2[x,y]/(x^a-1,y^b-1). This expression is closed-form and obtained directly from the corank of the module map; it does not require fitting or case-by-case selection. While concrete evaluation of the gcd employs the Euclidean algorithm, the formula itself remains input-independent in form. Existence conditions follow from algebraic constraints on the degree and support of this gcd, without enumeration of factor pairs beyond the ring structure. We will insert a clarifying sentence after the definition to emphasize this distinction. revision: partial

  2. Referee: The asymptotic-badness argument concludes that the family cannot be asymptotically good. The argument appears to rest on the dimension formula; if that formula is only computable per instance rather than closed-form, the uniform statement that the rate vanishes requires an additional uniform bound on the gcd degree that is not supplied.

    Authors: The vanishing-rate claim follows from the same explicit gcd-degree formula together with the observation that deg(gcd(f,g)) = o(ab) for all sequences with n=2ab→∞ under the polynomial families considered. This supplies the required uniform bound without needing per-instance computation. The argument is therefore independent of whether the gcd is evaluated algorithmically for specific examples. We will add a short lemma in the final section making the o(ab) bound explicit if the current exposition is deemed insufficiently uniform. revision: partial

Circularity Check

0 steps flagged

No circularity: dimension derived from independent algebraic structure of the bivariate polynomial ring

full rationale

The paper states it leverages the ring structure (F_2[x,y]/(x^a-1,y^b-1)) to predict dimension k and existence conditions. This is a standard algebraic coding-theory step: the quantum dimension equals twice the corank of the module map defined by the two bivariate polynomials, which equals 2(n - deg(gcd(f,g))) or equivalent Smith-form rank. The gcd or rank computation is a well-defined external algorithm on the input polynomials; it is not a fitted parameter, not renamed as a prediction, and not justified by self-citation. No load-bearing self-citation, no ansatz smuggled via prior work, and no self-definitional loop appear in the provided text. The derivation therefore supplies independent content rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities are described.

pith-pipeline@v0.9.0 · 5718 in / 965 out tokens · 20123 ms · 2026-05-23T02:44:49.412785+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

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

  1. Univariate Bicycle Quantum LDPC Codes: Explicit Logical Structure and Distance Bounds

    cs.IT 2026-05 unverdicted novelty 7.0

    Univariate bicycle codes give an explicit basis for logical operators and distance upper bounds in a restricted class of quantum LDPC codes while matching the performance of less constrained generalized and bivariate ...

  2. Symmetry-enriched topological order and quasifractonic behavior in $\mathbb{Z}_N$ stabilizer codes

    cond-mat.str-el 2025-11 unverdicted novelty 7.0

    Z_N bivariate-bicycle codes have essential topological properties determined by their Z_p prime-factor counterparts, enabling generalization of algebraic-geometric methods to anyon fusion rules and resolution of quasi...

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