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arxiv: 2508.19245 · v4 · submitted 2025-08-26 · 🪐 quant-ph · cond-mat.other· math-ph· math.MP

Composite-Dimensional Topological Codes with Boundaries and Defects

Mohamad Mousa , Amit Jamadagni , Eugene Dumitrescu This is my paper

Pith reviewed 2026-05-18 20:47 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.othermath-phmath.MP
keywords topological codesgapped boundariesdomain wallstwisted quantum doublescondensationquantum error correctionstabilizer modelsdefects
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The pith

Algorithms explicitly construct stabilizers for boundaries and defects in composite-dimensional twisted quantum doubles by condensing from the bulk.

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

The paper develops step-by-step algorithms to build stabilizer models for gapped boundaries, domain walls, and point defects in Abelian composite-dimensional twisted quantum doubles. It starts from the bulk model and uses condensation to derive the boundary stabilizers explicitly. This extends Pauli stabilizer models to describe non-translationally invariant topological orders. Examples include a new family of codes that couple the double of Z4 with the double semion phase. Threshold calculations using a composite dimensional belief propagation decoder compare their performance to surface codes.

Core claim

The authors introduce algorithms that construct boundary and domain-wall stabilizers starting from the bulk model using condensation for Abelian composite-dimensional twisted quantum doubles. This enables explicit microscopic stabilizer layouts for topological orders with boundaries and defects. The constructions are augmented by dimensional counting arguments and macroscopic pants decompositions that outline automation of code design, and the results are validated through error-correcting threshold calculations.

What carries the argument

The condensation procedure that derives explicit boundary and domain-wall stabilizers from the bulk stabilizer model by enforcing anyon condensations.

Load-bearing premise

The condensation procedure produces valid gapped stabilizer models when applied to composite-dimensional twisted doubles with boundaries and defects.

What would settle it

If simulations of the constructed codes reveal gapless modes or anyon statistics that mismatch the expected bulk topological order, the construction method would be falsified.

Figures

Figures reproduced from arXiv: 2508.19245 by Amit Jamadagni, Eugene Dumitrescu, Mohamad Mousa.

Figure 1
Figure 1. Figure 1: FIG. 1: Strings for creating e and m anyons in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: A logical-qubit [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Stabilizers layout for the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Left: a two-logical-qudit [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: A four-logical-qubit [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Ribbon operators for DS anyons. As discussed [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Stabilizers for the unique boundary of the DS [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: ) into a Z4 bulk. For simplicity, we now ignore, but return to in Sec. VI, the Z4 model’s taken exter￾nal boundary conditions (sphere, j-torus, smooth, rough, etc). The domain wall between the DS and Z4 condenses {1, 𝑒2𝑚2} of the Z4 anyons, while all anyons of DS can propagate into the Z4. In contrast to our prior exam￾ples, involving a single TQFT, the ℒZ4-DS domain wall is non-invertible. This implies th… view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Left: a logical-qubit DS- [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Comparison between [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Different condensation paths starting from a [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: After the fourth step of the algorithm, the sta [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: After the first two steps of the algorithm, the [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: After removing the qudits measured by the [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: The first three steps of the algorithm to con [PITH_FULL_IMAGE:figures/full_fig_p016_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: In the fourth step, the products of removed [PITH_FULL_IMAGE:figures/full_fig_p017_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: Steps five and six in the construction of the [PITH_FULL_IMAGE:figures/full_fig_p017_18.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20: In the fourth step, the products of removed [PITH_FULL_IMAGE:figures/full_fig_p018_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21: Steps five and six in the construction of the [PITH_FULL_IMAGE:figures/full_fig_p018_21.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23: Constructing a domain wall using condensa [PITH_FULL_IMAGE:figures/full_fig_p019_23.png] view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25: Top view of the two layers after the first two [PITH_FULL_IMAGE:figures/full_fig_p020_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: FIG. 26: After the third step, the stabilizers that do not [PITH_FULL_IMAGE:figures/full_fig_p020_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: FIG. 27: After the fourth step, products of old stabilizers [PITH_FULL_IMAGE:figures/full_fig_p021_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: FIG. 28: After the fifth step, the qudits in [PITH_FULL_IMAGE:figures/full_fig_p021_28.png] view at source ↗
Figure 30
Figure 30. Figure 30: FIG. 30: After the two steps of the algorithm, the edge [PITH_FULL_IMAGE:figures/full_fig_p021_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: FIG. 31: After the third step, old stabilizers that do not [PITH_FULL_IMAGE:figures/full_fig_p022_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: FIG. 32: In the fourth step, the products of removed [PITH_FULL_IMAGE:figures/full_fig_p022_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: , with one smooth and one rough boundary. FIG. 33: 𝐷(Z2) on a surface with rough and smooth boundaries meeting at only two points. Here, the number of rows and columns is three; 𝑁 = 3. Assuming that we have 𝑁 rows and 𝑁 columns, and noting that there are two edges per unit cell, the total Hilbert space dimension is 𝐷 = 2#𝐸 = 22𝑁2 . To count the number of generating stabilizer constraints, we also define t… view at source ↗
Figure 34
Figure 34. Figure 34: FIG. 34 [PITH_FULL_IMAGE:figures/full_fig_p024_34.png] view at source ↗
Figure 36
Figure 36. Figure 36: FIG. 36: Minimal stabilizer model for toric code. [PITH_FULL_IMAGE:figures/full_fig_p024_36.png] view at source ↗
Figure 38
Figure 38. Figure 38: FIG. 38: Stabilizers for the Toric code with two twists. [PITH_FULL_IMAGE:figures/full_fig_p025_38.png] view at source ↗
Figure 37
Figure 37. Figure 37: FIG. 37: Toric code with a single finite twist [PITH_FULL_IMAGE:figures/full_fig_p025_37.png] view at source ↗
Figure 39
Figure 39. Figure 39: FIG. 39: Stabilizer model for the [PITH_FULL_IMAGE:figures/full_fig_p025_39.png] view at source ↗
Figure 40
Figure 40. Figure 40: FIG. 40: Finite contractible DS patch embedded in a [PITH_FULL_IMAGE:figures/full_fig_p026_40.png] view at source ↗
Figure 41
Figure 41. Figure 41: FIG. 41: 2 contractible DS patches [PITH_FULL_IMAGE:figures/full_fig_p027_41.png] view at source ↗
Figure 44
Figure 44. Figure 44: FIG. 44: Left: the trivial domain wall denoted by [PITH_FULL_IMAGE:figures/full_fig_p028_44.png] view at source ↗
Figure 43
Figure 43. Figure 43: FIG. 43: Left: microscopic stabilizer model for the [PITH_FULL_IMAGE:figures/full_fig_p028_43.png] view at source ↗
Figure 45
Figure 45. Figure 45: FIG. 45: Any orientable 2D manifold can be decomposed [PITH_FULL_IMAGE:figures/full_fig_p029_45.png] view at source ↗
Figure 46
Figure 46. Figure 46: FIG. 46: Pants decomposition for the [PITH_FULL_IMAGE:figures/full_fig_p029_46.png] view at source ↗
Figure 47
Figure 47. Figure 47: FIG. 47: Pants decomposition of the torus with half [PITH_FULL_IMAGE:figures/full_fig_p030_47.png] view at source ↗
Figure 49
Figure 49. Figure 49: FIG. 49: Pants decomposition of the [PITH_FULL_IMAGE:figures/full_fig_p030_49.png] view at source ↗
Figure 50
Figure 50. Figure 50: FIG. 50: Pants decomposition of the [PITH_FULL_IMAGE:figures/full_fig_p031_50.png] view at source ↗
Figure 51
Figure 51. Figure 51: FIG. 51: Left: region of condensation [PITH_FULL_IMAGE:figures/full_fig_p032_51.png] view at source ↗
Figure 52
Figure 52. Figure 52: FIG. 52: After the third step, the stabilizers that do [PITH_FULL_IMAGE:figures/full_fig_p033_52.png] view at source ↗
Figure 54
Figure 54. Figure 54: FIG. 54: After stitching the stabilizers at the boundary [PITH_FULL_IMAGE:figures/full_fig_p034_54.png] view at source ↗
Figure 56
Figure 56. Figure 56: FIG. 56: After the third step, the stabilizers that did not [PITH_FULL_IMAGE:figures/full_fig_p035_56.png] view at source ↗
Figure 57
Figure 57. Figure 57: FIG. 57: The configuration after the fourth and fifth [PITH_FULL_IMAGE:figures/full_fig_p035_57.png] view at source ↗
Figure 58
Figure 58. Figure 58: FIG. 58: The final stabilizers for the defect after unfold [PITH_FULL_IMAGE:figures/full_fig_p036_58.png] view at source ↗
read the original abstract

We introduce new algorithms and provide example constructions of stabilizer models for the gapped boundaries, domain walls, and $0D$ defects of Abelian composite-dimensional twisted quantum doubles. Using the physically intuitive concept of condensation, our algorithm explicitly describes how to construct the boundary and domain-wall stabilizers starting from the bulk model. This extends the utility of Pauli stabilizer models in describing non-translationally invariant topological orders with gapped boundaries. To highlight this utility, we provide a series of examples, including a new family of quantum error-correcting codes where the double of $\mathbb{Z}_4$ is coupled to instances of the double semion (DS) phase. We discuss the codes' utility in the burgeoning area of quantum error correction with an emphasis on the interplay between deconfined anyons, logical operators, error rates, and decoding. We also augment our construction, built using algorithmic tools to describe the properties of explicit stabilizer layouts at the microscopic lattice-level, with dimensional counting arguments and macroscopic-level constructions building on pants decompositions. The latter outlines how such codes' representation and design can be automated. Our results are validated by a series of error-correcting threshold calculations comparing our code's performance with standard surface codes. To do so, we introduce a composite dimensional belief propagation decoder with ordered statistics that utilizes combination sweeps. Going beyond our worked-out examples, we expect our explicit step-by-step algorithms to pave the path for new higher-dimensional codes to be discovered and implemented in near-term architectures that take advantage of various hardware's distinct strengths.

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 introduces new algorithms for explicitly constructing stabilizer models for gapped boundaries, domain walls, and 0D defects of Abelian composite-dimensional twisted quantum doubles, starting from the bulk model via condensation. It supplies concrete example layouts, including a new family of codes coupling the Z_4 double to the double-semion phase, discusses utility for quantum error correction with emphasis on deconfined anyons, logical operators, and error rates, and validates performance via threshold calculations against surface codes using a newly introduced composite-dimensional belief propagation decoder with ordered statistics and combination sweeps. Additional macroscopic support is provided through dimensional counting arguments and pants decompositions for automated design.

Significance. If the constructions produce gapped Hamiltonians realizing the claimed topological orders, the work meaningfully extends Pauli stabilizer models to non-translationally invariant settings with boundaries and defects in composite dimensions. The explicit algorithmic recipes, the worked Z_4–double-semion example family, the threshold comparisons, and the specialized decoder constitute practical contributions that could aid near-term hardware implementations. The combination of microscopic lattice constructions with pants-decomposition automation arguments is a clear strength for reproducibility and extensibility.

major comments (2)
  1. [§3] §3 (condensation algorithm): The central claim that the condensation procedure yields gapped boundary and domain-wall stabilizers for composite-dimensional twisted doubles (e.g., Z_4 double coupled to double semion) is not supported by any explicit verification such as ground-state degeneracy counting, spectrum gap computation, or confirmation that the anyon content and logical operators match the condensed theory. Without such checks, the constructions risk producing gapless modes or inconsistent stabilizers due to the additional fusion/braiding constraints in composite dimensions.
  2. [§5.2] §5.2 (threshold calculations): The reported thresholds and performance comparisons to surface codes lack sufficient detail on Monte Carlo sample sizes, error-bar estimation, and data-exclusion criteria. This information is load-bearing for assessing whether the new codes genuinely outperform or match standard surface-code thresholds under the composite-dimensional decoder.
minor comments (2)
  1. [Abstract] Abstract: The phrase 'composite dimensional belief propagation decoder with ordered statistics that utilizes combination sweeps' is introduced without a one-sentence definition of its key novelty relative to standard BP decoders, which would improve accessibility.
  2. [Figure captions] Figure captions and lattice diagrams: Several example stabilizer layouts would benefit from explicit labeling of the condensed anyons or logical operators to make the microscopic-to-macroscopic correspondence immediate.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us improve the presentation and rigor of our results. We address each major comment in turn below.

read point-by-point responses

  1. Referee: [§3] §3 (condensation algorithm): The central claim that the condensation procedure yields gapped boundary and domain-wall stabilizers for composite-dimensional twisted doubles (e.g., Z_4 double coupled to double semion) is not supported by any explicit verification such as ground-state degeneracy counting, spectrum gap computation, or confirmation that the anyon content and logical operators match the condensed theory. Without such checks, the constructions risk producing gapless modes or inconsistent stabilizers due to the additional fusion/braiding constraints in composite dimensions.

    Authors: We agree that explicit verification strengthens the central claim, particularly given the additional fusion and braiding constraints present in composite dimensions. While the condensation procedure is constructed to preserve the topological order by design, we have added ground-state degeneracy counting for the Z_4–double-semion family in the revised manuscript; the computed degeneracies match the values predicted by the condensed theory. We have also included a table comparing the anyon content and logical operators extracted from the stabilizer models to those of the target condensed phase. Full spectrum gap computations remain computationally demanding for the system sizes considered, but we have added small-system exact-diagonalization results in the supplementary material that confirm a finite gap above the ground-state manifold. revision: yes

  2. Referee: [§5.2] §5.2 (threshold calculations): The reported thresholds and performance comparisons to surface codes lack sufficient detail on Monte Carlo sample sizes, error-bar estimation, and data-exclusion criteria. This information is load-bearing for assessing whether the new codes genuinely outperform or match standard surface-code thresholds under the composite-dimensional decoder.

    Authors: We concur that additional methodological details are necessary for a proper evaluation of the threshold results. In the revised manuscript we have expanded §5.2 to report the Monte Carlo sample sizes (ranging from 5×10^4 to 2×10^5 shots per data point, scaled with code distance), the error-bar estimation procedure (bootstrap resampling over independent decoder runs), and the data-exclusion criteria (discarding runs that failed to converge within a fixed iteration budget or exhibited outlier logical-error rates exceeding three standard deviations from the ensemble mean). These additions allow direct assessment of the statistical reliability of the reported thresholds. revision: yes

Circularity Check

0 steps flagged

No significant circularity; constructions derive from bulk models independently

full rationale

The paper's central contribution consists of explicit algorithmic constructions that build boundary, domain-wall, and defect stabilizers directly from a given bulk twisted quantum double model via condensation. These are supplemented by dimensional counting arguments and pants-decomposition macro-constructions, then validated through independent numerical threshold simulations and a newly introduced composite-dimensional belief-propagation decoder. No quoted step equates a derived quantity to a fitted parameter by construction, nor does any load-bearing premise reduce to a self-citation whose content is itself unverified within the paper. The derivation chain therefore remains self-contained against external benchmarks such as explicit lattice layouts and error-rate comparisons.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The constructions rest on the standard assumption that condensation produces valid gapped boundaries in twisted quantum doubles and that Pauli stabilizer models can describe the resulting non-translationally invariant orders. No explicit free parameters or invented entities are named in the abstract.

axioms (1)
  • domain assumption Condensation of anyons produces gapped boundaries and domain walls in Abelian twisted quantum doubles.
    Invoked when describing how to construct boundary stabilizers from the bulk model.

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