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arxiv: 2511.04430 · v2 · submitted 2025-11-06 · ❄️ cond-mat.str-el · math-ph· math.MP· quant-ph

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

Siyu He , Hao Song This is my paper

Pith reviewed 2026-05-18 00:57 UTC · model grok-4.3

classification ❄️ cond-mat.str-el math-phmath.MPquant-ph
keywords Z_N stabilizer codesbivariate-bicycle codessymmetry-enriched topological orderquasifractonic behavioranyon fusion rulesprime decompositionqudit codesstabilizer codes
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The pith

The essential topological properties of Z_N qudit stabilizer codes are fully determined by those of their Z_p prime qudit counterparts.

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

This paper studies a class of qudit stabilizer codes called Z_N bivariate-bicycle codes. The key result is that their main topological features, including anyon fusion and symmetry-enriched orders, can be found from the corresponding codes over prime dimensions p dividing N. This holds even when N has higher prime powers. By doing so, the work extends algebraic methods from prime qudits to general cases and explains quasifractonic behavior in models like the Delfino-Chamon-You code. An efficient algorithm using Gröbner bases computes the topological data and SET properties.

Core claim

From the polynomial representation of the Z_N bivariate-bicycle codes, the essential topological properties including anyon fusion rules and symmetry-enriched topological order can be determined by the properties of the Z_p counterparts for the prime factors p of N, even when N includes prime powers. This yields a simplification that allows generalization of algebraic-geometric methods such as the Bernstein-Khovanskii-Kushnirenko theorem to find anyon fusion rules in the general qudit case, and resolves the anyon mobility puzzle by elucidating the SET order in quasifractonic behavior.

What carries the argument

The polynomial representation of the Z_N bivariate-bicycle codes, which permits the topological data to factor through the prime decomposition of N.

If this is right

  • Anyon fusion rules for general N can be obtained by generalizing methods from prime dimensions.
  • The symmetry-enriched topological order explains the quasifractonic behavior and resolves anyon mobility issues in qudit BB codes.
  • An efficient Gröbner basis method over the integers computes both topological order and SET properties.
  • The framework leverages existing results on prime qudit codes for composite dimensions.

Where Pith is reading between the lines

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

  • Similar reductions might apply to other families of qudit codes beyond bivariate-bicycle.
  • This could simplify the design of topological quantum error correction codes with arbitrary qudit dimensions.
  • Experimental tests in qudit platforms could verify the predicted anyon behaviors for composite N.
  • Connections to modulated gauge theories may extend the simplification to other physical models.

Load-bearing premise

The polynomial representation of the Z_N codes allows topological data to factor completely through the prime decomposition without extra constraints from composite N.

What would settle it

Finding a specific Z_N bivariate-bicycle code for composite N where the anyon fusion rules or SET order cannot be predicted from the Z_p components alone.

Figures

Figures reproduced from arXiv: 2511.04430 by Hao Song, Siyu He.

Figure 1
Figure 1. Figure 1: FIG. 1. Illustration of the WCF model. (a) The [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Illustration of the DCY model. (a) the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The quasi-fractonic behavior of the DCY model. Here [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Illustration of the [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The topological index of the [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The topological indices of the [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The topological indices of the [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
read the original abstract

We study a broad class of qudit stabilizer codes, termed $\mathbb{Z}_N$ bivariate-bicycle (BB) codes, arising either as two-dimensional realizations of modulated gauge theories or as $\mathbb{Z}_N$ generalizations of binary BB codes. Our central finding, derived from the polynomial representation, is that the essential topological properties of these $\mathbb{Z}_N$ codes can be determined by the properties of their $\mathbb{Z}_p$ counterparts, where $p$ are the prime factors of $N$, even when $N$ contains prime powers ($N = \prod_i p_i^{k_i}$). This result yields a significant simplification by leveraging the well-studied framework of codes with prime qudit dimensions. In particular, this insight directly enables the generalization of the algebraic-geometric methods (e.g., the Bernstein-Khovanskii-Kushnirenko theorem) to determine anyon fusion rules in the general qudit situation. Moreover, we elucidate the symmetry-enriched topological (SET) order underlying the quasifractonic behavior in qudit BB codes (including the Delfino-Chamon-You model), resolving the associated anyon mobility puzzle. We also develop an efficient computational algebraic method, based on Gr\"{o}bner bases over the ring of integers, to determine both the topological order and its SET properties.

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

3 major / 2 minor

Summary. The manuscript studies a class of qudit stabilizer codes called Z_N bivariate-bicycle (BB) codes, realized either as modulated gauge theories or as generalizations of binary BB codes. Its central claim, derived from the polynomial representation of the stabilizers, is that the essential topological properties—including anyon fusion rules, symmetry-enriched topological (SET) order, and quasifractonic mobility—can be completely determined from the corresponding Z_p codes for the prime factors p of N, even when N contains prime powers (N = ∏ p_i^{k_i}). The work generalizes algebraic-geometric tools such as the Bernstein-Khovanskii-Kushnirenko theorem to the composite-qudit setting, resolves the anyon-mobility puzzle in models including the Delfino-Chamon-You code, and introduces an efficient Gröbner-basis algorithm over the ring of integers to compute both the topological order and its SET properties.

Significance. If the reduction through prime factors holds without loss of topological invariants, the result would substantially simplify the analysis of general-qudit topological codes by allowing reuse of the well-developed Z_p framework. The integer Gröbner-basis method and the explicit treatment of SET order in qudit BB codes constitute concrete computational and conceptual advances. The paper supplies machine-checkable algebraic machinery and falsifiable predictions for anyon content and mobility, which strengthens its utility for the field.

major comments (3)
  1. Abstract and the section introducing the polynomial representation: the central claim that anyon fusion rules and SET data factor exactly through the Z_p reductions (even for N = p^k) is load-bearing, yet the manuscript must explicitly verify that the stabilizer module over Z_N[x^{±1}, y^{±1}] acquires no additional p-torsion or annihilators under Gröbner reduction that would alter the fusion ring or symmetry action on anyons. A concrete example for N = p^2 (e.g., explicit primary decomposition or Chinese-Remainder lifting) is required to confirm that the topological invariants match those obtained from the separate Z_p computation.
  2. Section on algebraic-geometric methods and the BKK generalization: the extension of the Bernstein-Khovanskii-Kushnirenko theorem to Z_N is presented as resolving the composite case, but the argument should include a step-by-step check that the zero-dimensional ideal and its multiplicity data remain unchanged under the prime-power decomposition; otherwise the claimed simplification for fusion rules does not follow.
  3. Section on quasifractonic behavior and the Delfino-Chamon-You model: the resolution of the anyon-mobility puzzle via SET order relies on the same reduction; the manuscript should demonstrate that the symmetry action on anyons (including mobility constraints) is preserved exactly when lifting from Z_p to Z_{p^k}, with no extra relations introduced by the non-integral-domain structure of Z_N.
minor comments (2)
  1. Notation: the distinction between the Laurent polynomial ring over Z_N versus over Z should be stated more explicitly when introducing the stabilizer module, to avoid ambiguity in the Gröbner-basis computations.
  2. The abstract asserts that the central finding 'follows from the polynomial representation' without supplying derivation steps; a brief outline of the key algebraic steps in the introduction would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments identify key points where additional explicit verification would strengthen the presentation of our reduction from Z_N to Z_p. We respond to each major comment below and will incorporate the requested checks in the revised manuscript.

read point-by-point responses

  1. Referee: Abstract and the section introducing the polynomial representation: the central claim that anyon fusion rules and SET data factor exactly through the Z_p reductions (even for N = p^k) is load-bearing, yet the manuscript must explicitly verify that the stabilizer module over Z_N[x^{±1}, y^{±1}] acquires no additional p-torsion or annihilators under Gröbner reduction that would alter the fusion ring or symmetry action on anyons. A concrete example for N = p^2 (e.g., explicit primary decomposition or Chinese-Remainder lifting) is required to confirm that the topological invariants match those obtained from the separate Z_p computation.

    Authors: We agree that an explicit verification for prime-power cases strengthens the central claim. In the revised manuscript we will add a concrete example for N = p^2 (e.g., N=4), performing primary decomposition of the stabilizer ideal over Z_N[x^{±1}, y^{±1}] and applying the Chinese Remainder Theorem to lift from the Z_p factors. This computation will show that no additional p-torsion or annihilators arise that modify the fusion ring or anyon symmetry actions, confirming exact agreement with the separate Z_p results. revision: yes

  2. Referee: Section on algebraic-geometric methods and the BKK generalization: the extension of the Bernstein-Khovanskii-Kushnirenko theorem to Z_N is presented as resolving the composite case, but the argument should include a step-by-step check that the zero-dimensional ideal and its multiplicity data remain unchanged under the prime-power decomposition; otherwise the claimed simplification for fusion rules does not follow.

    Authors: We will expand the algebraic-geometric methods section with a step-by-step verification. Starting from the ideal over Z_N, we will explicitly reduce via the prime-power factorization, compute the zero-dimensional ideal and its multiplicity data at each stage, and confirm that these quantities are invariant under the decomposition. This establishes that the BKK generalization carries over without loss, supporting the simplification for anyon fusion rules. revision: yes

  3. Referee: Section on quasifractonic behavior and the Delfino-Chamon-You model: the resolution of the anyon-mobility puzzle via SET order relies on the same reduction; the manuscript should demonstrate that the symmetry action on anyons (including mobility constraints) is preserved exactly when lifting from Z_p to Z_{p^k}, with no extra relations introduced by the non-integral-domain structure of Z_N.

    Authors: We will add an explicit demonstration in the quasifractonic behavior section for the Delfino-Chamon-You model. By direct computation of the symmetry action on anyons when lifting from Z_p to Z_{p^k}, we will verify that mobility constraints and SET data are preserved exactly, with no additional relations arising from the non-integral-domain structure of Z_N. This confirms that the SET resolution of the mobility puzzle holds for composite N. revision: yes

Circularity Check

0 steps flagged

Reduction to Z_p via polynomial representation is algebraically independent, not circular

full rationale

The central claim follows from the standard bivariate polynomial encoding of stabilizer modules over the Laurent polynomial ring with coefficients in Z_N. The paper applies Gröbner-basis reduction over Z and the Bernstein-Khovanskii-Kushnirenko theorem (generalized to the qudit setting) to extract anyon fusion rules and SET data; these are external algebraic tools whose correctness does not depend on the target result. For composite N the reduction to prime-power factors is obtained by the Chinese-Remainder decomposition of the coefficient ring together with primary decomposition of the module, which is a ring-theoretic fact rather than a fitted or self-referential definition. No load-bearing step reduces to a self-citation, an ansatz smuggled from prior work, or a parameter fitted to the very quantities being predicted. The computational method supplies an independent verification route, keeping the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; the central claim rests on the assumption that the polynomial representation of the codes factors through the prime decomposition of N. No explicit free parameters, invented entities, or additional axioms are stated.

axioms (1)
  • domain assumption Topological properties of Z_N bivariate-bicycle codes are captured by a polynomial representation that factors over the prime divisors of N
    This is the explicit basis for the reduction claim stated in the abstract.

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