{"id":"62805d03-82e3-4ed9-aefe-703f43064d01","arxiv_id":"2610.36029","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"AlgebrasTQD computes all condensable algebras, subalgebra relations, Hasse diagrams, and phase types for twisted quantum doubles D^ω(G) of any finite group G and 3-cocycle ω.","lead":"AlgebrasTQD is an open-source GAP code that computes all condensable algebras and topological interfaces for twisted quantum doubles of any finite group with any 3-cocycle anomaly. It enables classification of 1+1d gapped and gapless phases, anyon proliferation transitions, and construction of non-Clifford logical gates for fault-tolerant quantum computing.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"No significant objection identified: the central claim is well-supported by a known classification theorem, the algorithmic reductions are proven in the appendix, and all examples match prior results.","rationale":"The reader's weakest assumption—the completeness of the (H,F,γ,ε) classification—is indeed the load-bearing premise for the code's claim to compute every condensable algebra. However, this is a known theorem from [30,31], not a gap in the paper's own reasoning. The paper's contributions lie in efficiently implementing that classification, providing proofs of algorithmic optimizations (Appendix A), and cross-checking against known examples. The code is open-source and reproduceable. The paper honestly states that the modular data alone does not determine the theory but the code uses the full group and cocycle data, mitigating that risk. I find no additional load-bearing concern beyond the acknowledged premise, and the reader's ACCEPT verdict is justified.","tokens_in":42361,"tokens_out":2017,"duration_ms":19106,"concrete_test":"Independently run AlgebrasTQD on a representative set of groups (e.g., Z_2^3, D_8, A_4, SmallGroup(32,43)) and verify that: (1) the output list of algebras matches those reported in the paper's tables, (2) the number of isomorphism classes equals the number of distinct anyon decompositions plus identified twin pairs, and (3) every Lagrangian algebra A(H,γ) has dim = |G|. The paper's example scripts already perform these checks.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim—that AlgebrasTQD computes all condensable algebras for any finite group G and 3-cocycle ω—rests on the classification theorem of [30,31] stating that every condensable algebra in Z(Vec^ω G) is of the form A(H,F,γ,ε). This is a published, peer-reviewed result, and the paper does not attempt to reprove it. The algorithmic steps for finding γ, ε, isomorphism classes, and inclusions are rigorously justified in Appendix A (Theorems A.1, A.3, Lemma A.2, Lemma A.4). The code also verifies its own outputs with internal consistency checks (e.g., non-negative integer multiplicities, cocycle conditions). The examples reproduce the known algebra counts for D(S_3), D(D_8), D(A_4), and groups of order 32 and 48, including twin pairs. No internal inconsistency, missing case, or reliance on an unverified premise is identified. The acknowledged limitation—that the classification theorem is taken as input—is both proper and low-risk given the theorem's standing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper introduces AlgebrasTQD, an open-source GAP code that computes condensable algebras (equivalently, topological boundaries and interfaces) in twisted quantum doubles D^ω(G) for any finite group G and 3-cocycle ω.  The code implements the classification of condensable algebras as A(H,F,γ,ε) from the work of Davydov and Simmons; it generates all such data structures, decomposes them into anyons, sorts them into isomorphism classes (including twin algebras), computes subalgebra relations and Hasse diagrams, determines the associated 1+1d gapped/gapless phases, constructs the reduced topological order and the anyon map across interfaces, and returns non-Clifford logical gates for non-Abelian surface codes.  The paper states the mathematical formulas, proves the algorithmic optimizations in Appendix A, and illustrates the code with a series of examples (Z_2^3 with type-III cocycle, D_8, A_4 with and without anomaly, twin algebras in a group of order 32 and in GL(2,3), and Clifford-hierarchy gates for dihedral groups).  All calculations are exact and verified by internal consistency checks.","tokens_in":42702,"tokens_out":2802,"duration_ms":28990,"significance":"If the central claim holds—that the code correctly implements the known classification theorem and produces all condensable algebras for any finite group and cocycle—the paper provides a substantial computational tool for several active research areas: the categorical Landau paradigm for 1+1d phases, anyon condensation and proliferation in 2+1d topological orders, and the construction of non-Clifford logical gates in topological quantum codes.  The code automates a calculation that quickly becomes impractical by hand (e.g., groups of order 32 already have hundreds of algebras), and it includes the identification of twin algebras and phases that have only recently been recognized.  The examples reproduce known results (38 algebras for D(D_8), 517 classes for G32, etc.) and thus inspire confidence.  The mathematical optimizations are proved in an appendix, and the code performs its own checking of cocycle conditions, integrality of multiplicities, and dimension sums.","major_comments":[{"comment":"The paper's central claim—that AlgebrasTQD computes 'all condensable algebras'—rests on the completeness of the classification theorem [30,31] that every condensable algebra in Z(Vec^ω G) is of the form A(H,F,γ,ε). The authors acknowledge this dependence (e.g., §3.2, final paragraph of §2.4, and the reader's note). No internal inconsistency or omitted case is identified. I therefore raise no major objections on this point. The algorithmic proofs in Appendix A are rigorous and the output verifications are systematic.","section":"Entire manuscript (scope statement)","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The abstract and introduction state that the code 'computes all condensable algebras'. It might be helpful to add an explicit sentence in the Introduction (or at the end of §3.2) stating that completeness is conditional on the classification of [30,31] and that the code searches only over the data (H,F,γ,ε) that that theorem provides. The limitation is already stated in the text but could be made more prominent for newcomers.","section":"§1 (Introduction)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The list of 38 algebras for D(D_8) is printed with many entries labelled by '1+e_G' etc. Although the notation is explained in the surrounding text and in the discussion of Z_2^3, a brief caption or a legend in the printed output (or in the description of the example) would improve readability for readers who jump directly to this section.","section":"§5.4 (Condensable Algebras)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The table caption says 'canonical interface maps', but the text indicates that the map is not unique (it is one of several, and the code can generate all with AllInterfaceMaps). Please clarify in the caption that the table shows one representative map for each algebra.","section":"Table 3","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The code computes logical gates for D_4, D_8, and D_16 (N=1,2,4). The relationship between the group order and the gate T^(1/N) is explained, but a brief justification of why the level in the Clifford hierarchy is (2N+1) could be added for completeness (the current text only reports the numerical output).","section":"§5.9 (Quantum Computing Gates)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"In the multiplication relation for twisted-sector operators, the phase γ(f_1,f_2) appears. Below in Eq. (12) there is an additional factor Ω_r(f,f'). It might be helpful to note explicitly that Ω_r is defined in Eq. (3) and that the two equations are consistent because the operator algebra is the full algebra structure, not only the multiplication of the O_f.","section":"§3.2, Eq. (9)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The source file '0_Modular_Data_Mignard-Schauenburg-Gruen-Morrison.g' is credited to other authors. The paper states that its copyright remains with its authors and that it is part of the TeX source of [104]. This is a proper acknowledgment. A short note on how to cite that external work when using AlgebrasTQD for a research project would be helpful to users.","section":"§4.2 (Source Files)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"In Theorem A.1, the statement that 'ω is a coboundary' is used to assert existence of a normalized γ. The proof correctly constructs the solution via the linear solver. The condition that ω|_F is a coboundary is later checked by the code (GaugeableSubgroups filters subgroups on which ω is trivial). This is logically sound. A minor typographical issue: in the proof, the line 'γ = γ'/κ' should clarify that κ is the constant 2-cochain with value κ = γ'(1,1). The current text says 'κ = γ'(1,1)' and 'dκ=1', which is fine; adding 'constant' would eliminate any ambiguity.","section":"Appendix A","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The conclusion mentions extensions to higher dimensions and quantum error correction. These are natural directions. The paper does not claim to implement them. The forward-looking tone is appropriate.","section":"§6 (Conclusion and Outlook)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The paper uses both the notation D_8 (order 8) and D_4N (order 8N). In §5.9, the dihedral group of order 8 is called D_4 following the references [22,25]; earlier in §5.2 it is called D_8. The switch in notation could confuse a reader skimming. A brief footnote or a remark clarifying that D_4 in §5.9 is the same group as D_8 from earlier would be helpful.","section":"General","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"The paper presents AlgebrasTQD, a GAP package that computes condensable algebras in twisted quantum doubles. The main thing to know: this is an engineering and documentation contribution, not a mathematical one. The code implements the classification of algebras by (H,F,γ,ε) from work by Davydov, Simmons, Gai, Schäfer-Nameki, and the author herself. All the heavy lifting—exact linear algebra over roots of unity, solving for gamma and epsilon, determining isomorphism classes, Hasse diagrams, and interface maps—is done carefully. The appendix proves the key algorithmic reductions (Theorems A.1, A.3, Lemmas A.2, A.4). The examples are thorough: they reproduce known counts for D(S3), D(D8), D(A4), and groups of order 32 and 48, including twin algebras. The code ships with its dependencies and runs in GAP. That is real value: researchers who need to enumerate algebras for a given (G,ω) no longer have to reimplement the linear algebra by hand.\n\nThe paper is honest about what it takes as input. The classification theorem from [30,31] is cited and not reproved. The authors state clearly that if a condensable algebra exists outside that classification, the code would miss it. That is an acknowledged limitation and not a flaw in the paper itself—the paper is a wrapper around a theorem. The code internally checks all consistency conditions (non-negative integer multiplicities, dimension sums, cocycle conditions), so you get a guarantee that the output satisfies the algebra axioms.\n\nSoft spots: The paper is long (38 pages plus appendix and references), partly because every example is spelled out in text. That is a feature for a user manual but makes the paper feel like a tutorial. The quantum computing section (Section 5.9, logical gates from dihedral groups) is a brief illustration rather than a full treatment—fine for an outlook but not a core part of the contribution. The citation list leans heavily on the author's own work (co-authored with Gai and Schäfer-Nameki), but that is appropriate because the code grew out of those projects. No citation is missing or misrepresented.\n\nWho is this for? Anyone working on topological phases with finite group symmetry, especially in the SymTFT / generalized symmetries community. If you need to compute algebra data for a given group, this is the tool. It deserves a serious referee. I would bring it to reading group and would cite it.","headline":"Solid computational tool that faithfully implements known mathematics; no new results but thoroughly done and clearly useful.","tokens_in":43163,"tokens_out":1048,"would_cite":true,"duration_ms":12408,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"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…","keywords":["condensable algebra","twisted quantum double","Dijkgraaf-Witten theory","anyon condensation","symmetry topological field theory","non-Abelian surface code","non-Clifford gate","twin algebras"],"falsifier":"Exhibit a twisted quantum double D^ω(G) that has a condensable algebra whose simple object decomposition cannot be realised by any pair (H,F) of subgroups with admissible γ and ε, either by a direct counterexample or by proving that the classification theorem itself has a missing case.","tokens_in":43865,"feed_emoji":"🌀","tokens_out":3028,"duration_ms":30024,"temperature":0.7,"pith_summary":"AlgebrasTQD claims to compute every condensable algebra (also called connected étale algebra) in the twisted quantum double D^ω(G), the symmetry TFT for 1+1d systems with finite group symmetry G and anomaly ω. The code is built on the group-theoretical classification of such algebras by the data (H,F,γ,ε), and it returns the full algebra structure, the anyon decomposition, the isomorphism classes (including twin algebras with the same anyons), the subalgebra lattice and its Hasse diagram, the reduced topological order for non-maximal algebras, the map of anyons across the associated topological interface, and the logical gates for non-Abelian surface codes derived from certain dihedral groups. A sympathetic reader would care because this automates a core computation that was previously done by hand or for small groups, enabling systematic study of gapped and gapless phases with non-invertible symmetries, of anyon proliferation transitions, and of fault-tolerant quantum computing with non-Abelian anyons.","feed_headline":"AlgebrasTQD computes all condensable algebras in twisted quantum doubles","feed_subtitle":"New GAP code enables classification of 1+1d phases and non-Clifford gates via anyon condensation.","key_machinery":"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.","core_discovery":"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}).","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provides the classification of condensable algebras in group-theoretical modular categories by the data (H,F,γ,ε), which is the mathematical foundation of the entire code.","marker":"[30]"},{"why":"Proves the classification of Lagrangian algebras in group-theoretical braided fusion categories, giving the rigorous basis for the algebra construction and the conditions (22).","marker":"[31]"},{"why":"Introduces twin algebras, the formula for anyon decomposition (14), and the isomorphism tests (15); the code implements these to find twin pairs.","marker":"[32]"},{"why":"Establishes the Hasse diagram and phase classification (gapped and gapless) from condensable algebras with non-invertible symmetries; the code reproduces and extends those phase labels.","marker":"[11]"},{"why":"Supplies the modular data (S and T matrices) of pointed fusion categories and projective characters of centralisers, used by TQDTheory to set up the anyon theory.","marker":"[104]"},{"why":"Describes the hybrid lattice surgery protocol for non-Clifford gates that the code implements in D4NSurgeryGate.","marker":"[22]"},{"why":"GAP is the platform and language in which AlgebrasTQD is written; the code depends on GAP's group-theory and homological-algebra packages.","marker":"[34]"}],"fun_headline_variants":["GAP code maps anyon condensation in twisted quantum doubles","AlgebrasTQD: classify phases and build non-Clifford gates","Open-source tool computes all condensable algebras in twists","Non-Abelian anyon condensation and logical gates in GAP","Exact classification of topological phases via AlgebrasTQD"],"cache_read_input_tokens":0,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["GAP code maps anyon condensation in twisted quantum doubles","AlgebrasTQD: classify phases and build non-Clifford gates","Open-source tool computes all condensable algebras in twists","Non-Abelian anyon condensation and logical gates in GAP","Exact classification of topological phases via AlgebrasTQD"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":7.5e-05,"raw_usage":{"cost":7.49952e-05,"is_byok":false,"cost_details":{"upstream_inference_cost":7.49952e-05,"upstream_inference_prompt_cost":4.51584e-05,"upstream_inference_completions_cost":2.98368e-05},"total_tokens":1490,"prompt_tokens":1120,"completion_tokens":370,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":0,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":284}},"tokens_in":1120,"tokens_out":370,"cost_source":"openrouter_usage","duration_ms":4748,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":284,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-28T17:12:30.430574+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Exhibit a twisted quantum double D^ω(G) that has a condensable algebra whose simple object decomposition cannot be realised by any pair (H,F) of subgroups with admissible γ and ε, either by a direct counterexample or by proving that the classification theorem itself has a missing case.","supporting_citations":[{"cited_title":"Twin Algebras: Condensable Algebras beyond Anyons","cited_arxiv_id":"2605.31602","evidence_quote":"Introduces twin algebras, the formula for anyon decomposition (14), and the isomorphism tests (15); the code implements these to find twin pairs."},{"cited_title":"Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries,","cited_arxiv_id":null,"evidence_quote":"Establishes the Hasse diagram and phase classification (gapped and gapless) from condensable algebras with non-invertible symmetries; the code reproduces and extends those phase labels."},{"cited_title":"Computing Modular Data for Pointed Fusion Categories","cited_arxiv_id":"1808.05060","evidence_quote":"Supplies the modular data (S and T matrices) of pointed fusion categories and projective characters of centralisers, used by TQDTheory to set up the anyon theory."},{"cited_title":"Hybrid Lattice Surgery: Non-Clifford Gates via Non-Abelian Surface Codes","cited_arxiv_id":"2510.20890","evidence_quote":"Describes the hybrid lattice surgery protocol for non-Clifford gates that the code implements in D4NSurgeryGate."}],"review_version":1}