REVIEW
AlgebrasTQD: Computing Algebras and Interfaces in Twisted Quantum Doubles
Strongly Correlated Electrons · Submitted 26 Oct 2026 · 138 references
Pith's verdict: Solid computational tool that faithfully implements known mathematics; no new results but thoroughly done and clearly useful.
T0 · AI review
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Referee report
The paper
The paper claims
AlgebrasTQD is a GAP code that computes all condensable algebras in any twisted quantum double D^ω(G) for any finite group G and 3-cocycle ω, including their anyon decomposition, isomorphism classes, Hasse diagrams, reduced topological…
The reading
What carries the argument
The central object is the group-theoretical data (H,F,γ,ε) that classifies condensable algebras in Z(Vec^ω_G): H ≤ G is the preserved subgroup, F ◁ H is the subgroup of twisted sectors, γ: F×F → U(1) trivialises ω|_F, and ε: H×F → U(1) satisfies the consistency relations of Davydov–Simmons. The code reduces all computations—finding γ, ε, isomorphism classes, inclusions, and the twist α of the reduced topological order—to solving linear systems over Q/Z using the Hermite normal form of an integer matrix, applied to equations on generating sets rather than the whole group.
Extended reading notes
Core claim
The paper presents an open-source GAP package that implements the classification of condensable algebras in Z(Vec^ω_G) (the Drinfeld center of Vec^ω_G) via the data A(H,F,γ,ε), computes their anyon multiplicities using an algebraic formula, and organises them into isomorphism classes, subalgebra inclusions, and phase types. It also computes the 3-cocycle of the reduced topological order D^α(H/F) and the map of anyons across the interface. The code is exact and works for any finite group and 3-cocycle, limited only by time and group size. The paper demonstrates the code on several examples: the toric code, double semion, D(S_3), D(D_8), D(A_4) with and without twists, the twin algebras of G_32 and GL(2,3), and the non-Clifford gates from D(D_{4N}).
Load-bearing premise
The classification of condensable algebras in twisted quantum doubles by the group-theoretical data (H,F,γ,ε) is assumed to be complete; if there exists a condensable algebra that cannot be described by such data, the code will not find it.
Editorial extensions
If this is right
- The code classifies all 1+1d gapped and gapless phases with symmetry group G and anomaly ω (or any symmetry obtained by gauging) by computing every condensable algebra and their Hasse diagram, making the categorical Landau paradigm concrete for any finite group.
- Twin algebras—inequivalent algebras with the same anyon decomposition—can be systematically found; these correspond to phases that are intrinsically deconfined quantum critical points in 1+1d, beyond any Landau description.
- Non-maximal algebras yield the reduced topological order and the explicit map of anyons across the interface, enabling the construction of new phase transitions from simpler input transitions via the 'club sandwich' SymTFT.
- Non-Clifford logical gates at every level of the Clifford hierarchy can be constructed from non-Abelian surface codes using the code's functions for hybrid lattice surgery and SPT stacking.
- The exact linear solver reduces the complexity of finding γ and ε from solving equations on all group elements to solving on generating sets, making computations feasible for groups of order up to at least 48.
Reading between the lines
- The code could be extended to compute condensable algebras in (3+1)d Dijkgraaf–Witten theories (the SymTFT for 2+1d symmetries) once the analogous group-theoretical data for ‘minimal algebras’ is understood, which the author points out as future work.
- With the code, one can in principle scan all finite groups up to a given order to find twin algebras and twin phases, mapping out the full landscape of intrinsic deconfined critical points for group-theoretical symmetries.
- The optimisation that reduces the ε equations to a generating set (Theorem A.3) could be adapted to other contexts where cohomological data is needed, such as computing symmetry fractionalisation classes in topological orders.
- The combination of algebra data with tensor-network methods would enable numerical simulation of the lattice models for twin phases and anyon proliferation transitions beyond exact diagonalisation.
Original abstract
We present AlgebrasTQD, an open-source code written in GAP (Groups, Algorithms, and Programming), for computing condensable algebras and the corresponding topological boundaries and interfaces of twisted quantum doubles $D^\omega(G)$. These algebras can be used to classify 1+1d gapped and gapless phases whose symmetry is related by gauging to a finite group $G$ with anomaly $\omega\in H^3(G,U(1))$, to study 2+1d anyon proliferation and to construct logical gates in non-Abelian topological quantum codes. The functions in AlgebrasTQD compute all condensable algebras $A(H,F,\gamma,\epsilon)$ with their algebra structure and anyons, sort them into isomorphism classes and identify the twin algebras that are inequivalent but share the same anyon decomposition. AlgebrasTQD also determines all the subalgebra relations, the corresponding Hasse diagram and the types of 1+1d phases for any chosen, possibly non-invertible, group-theoretical symmetry. For non-maximal algebras, the code computes the reduced topological order with its $3$-cocycle and the map of anyons across the topological interface. Finally, AlgebrasTQD includes functions that construct certain non-Clifford logical gates via non-Abelian surface codes. We summarize the formulas that are implemented in the AlgebrasTQD code, and we illustrate its use with Abelian and non-Abelian groups including $\mathbb{Z}_2^3$ with $\omega_\text{III}$, $D_8$, $A_4$ with and without anomaly, twin algebras and twin phases, and the Clifford-hierarchy gates in quantum doubles of dihedral groups. All arithmetic is exact and AlgebrasTQD can be used with any finite group $G$ and 3-cocycle $\omega$, limited only by the time and size of the calculation. The code is available at https://github.com/AlisonWarman/AlgebrasTQD
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We present AlgebrasTQD, an open-source code written in GAP (Groups, Algorithms, and Programming), for computing condensable algebras and the corresponding topological boundaries and interfaces of twisted quantum doubles $D^\omega(G)$. These algebras can be used to classify 1+1d gapped and gapless phases whose symmetry is related by gauging to a finite group $G$ with anomaly $\omega\in H^3(G,U(1))$, to study 2+1d anyon proliferation and to construct logical gates in non-Abelian topological quantum codes. The functions in AlgebrasTQD compute all condensable algebras $A(H,F,\gamma,\epsilon)$ with their algebra structure and anyons, sort them into isomorphism classes and identify the twin algebras that are inequivalent but share the same anyon decomposition. AlgebrasTQD also determines all the subalgebra relations, the corresponding Hasse diagram and the types of 1+1d phases for any chosen, possibly non-invertible, group-theoretical symmetry. For non-maximal algebras, the code computes the reduced topological order with its $3$-cocycle and the map of anyons across the topological interface. Finally, AlgebrasTQD includes functions that construct certain non-Clifford logical gates via non-Abelian surface codes. We summarize the formulas that are implemented in the AlgebrasTQD code, and we illustrate its use with Abelian and non-Abelian groups including $\mathbb{Z}_2^3$ with $\omega_\text{III}$, $D_8$, $A_4$ with and without anomaly, twin algebras and twin phases, and the Clifford-hierarchy gates in quantum doubles of dihedral groups. All arithmetic is exact and AlgebrasTQD can be used with any finite group $G$ and 3-cocycle $\omega$, limited only by the time and size of the calculation. The code is available at https://github.com/AlisonWarman/AlgebrasTQD
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We present AlgebrasTQD, an open-source code written in GAP (Groups, Algorithms, and Programming), for computing condensable algebras and the corresponding topological boundaries and interfaces of twisted quantum doubles $D^\omega(G)$. | contribution 0.97 | 0.92 | 0.23 | asserted 1.00 |
| s1 | These algebras can be used to classify 1+1d gapped and gapless phases whose symmetry is related by gauging to a finite group $G$ with anomaly $\omega\in H^3(G,U(1))$, to study 2+1d anyon proliferation and to construct logical gates in non-Abelian topological quantum codes. | interpretation 0.59 | 0.25 | 0.15 | asserted 0.77 |
| s2 | The functions in AlgebrasTQD compute all condensable algebras $A(H,F,\gamma,\epsilon)$ with their algebra structure and anyons, sort them into isomorphism classes and identify the twin algebras that are inequivalent but share the same anyon decomposition. | method 0.64 | 0.87 | 0.10 | asserted 1.00 |
| s3 | AlgebrasTQD also determines all the subalgebra relations, the corresponding Hasse diagram and the types of 1+1d phases for any chosen, possibly non-invertible, group-theoretical symmetry. | contribution 0.68 | 0.86 | 0.08 | asserted 1.00 |
| s4 | For non-maximal algebras, the code computes the reduced topological order with its $3$-cocycle and the map of anyons across the topological interface. | contribution 0.60 | 0.83 | 0.05 | asserted 1.00 |
| s5 | Finally, AlgebrasTQD includes functions that construct certain non-Clifford logical gates via non-Abelian surface codes. | contribution 0.94 | 0.85 | 0.07 | asserted 0.98 |
| s6 | We summarize the formulas that are implemented in the AlgebrasTQD code, and we illustrate its use with Abelian and non-Abelian groups including $\mathbb{Z}_2^3$ with $\omega_\text{III}$, $D_8$, $A_4$ with and without anomaly, twin algebras and twin phases, and the Clifford-hierarchy gates in quantum doubles of dihedral groups. | contribution 0.77 | 0.89 | 0.13 | asserted 1.00 |
| s7 | All arithmetic is exact and AlgebrasTQD can be used with any finite group $G$ and 3-cocycle $\omega$, limited only by the time and size of the calculation. | limitation 0.59 | 0.71 | 0.09 | asserted 0.99 |
| s8 | The code is available at https://github.com/AlisonWarman/AlgebrasTQD | contribution 0.61 | 0.44 | 0.03 | asserted 0.98 |
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Figures
Figures reproduced from arXiv:2610.36029 by the authors.
References
Discussion
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