{"id":"cd27c896-0917-4610-9ee2-04568fdf946e","arxiv_id":"2608.09915","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Every 2D translation-invariant topological CSS code is local-unitarily equivalent, at the maximal anyon-preserving translation symmetry, to stacks of toric codes and product states, with an explicit polynomial-time construction.","lead":"This paper proves that every 2D translation-invariant topological CSS quantum error-correcting code can be mapped, by a local quantum circuit, to a stack of simple toric codes and product states while keeping as much translation symmetry as possible. It also gives a polynomial-time algorithm to build the map, with bounds on the supercell size and the spreading of operators.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Lemma 6 is compressed but its full-rank claim is correct.","rationale":"The reader identified Lemma 6 as the weakest assumption, and I agree that it is the most load-bearing step in the proof: without full rank of the extension-class representative, the GL(r,F2) transitivity argument cannot reach the standard extension and the decoupling theorem would fail. However, the concern does not land as a correctness defect. The full-rank assertion follows from a standard Tor argument: tensoring the input short exact sequence with the residue field R/J yields a left-derived five-term exact sequence whose connecting map is the extension-class representative; exactness makes that map injective. This both explains why the middle module being free forces full rank and supplies the missing justification for the compressed reference-extension classification. The only other noticed issue is a remark whose stated determinant criterion for lifting F2-invertible matrices is not valid for arbitrary lifts in a Laurent polynomial ring, but the needed lifting is true by decomposing GL(n,F2) into elementary matrices and lifting those, so Lemma 5's lift step remains sound. The complexity estimates are conservative but polynomial, and the numerical and independent-work support corroborate the central claim. I therefore recommend no change to the reader's ACCEPT verdict.","tokens_in":38976,"tokens_out":48987,"duration_ms":472104,"concrete_test":"Verify Lemma 6 independently by computing the connecting homomorphism δ in the long exact sequence obtained by tensoring the short exact sequence (IV.9) with R/J for a nontrivial example, e.g., the coarse-grained 6.6.6 color code with t=2. Concretely, compute the extension-class matrix from the standard resolution of J^t using a Gröbner-basis presentation, then check rank(δ)=t. Repeat for a few random small exact CSS complexes; any rank-deficient δ would contradict exactness and falsify the claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single load-bearing point is Lemma 6: existence of the chain isomorphism requires the extension-class representative of the input complex to have full rank t. The proof in Supplemental Sec. IV.C classifies reference extensions and asserts that a free middle module forces full rank, but the argument is abbreviated. I checked this assertion directly. Tensoring the short exact sequence (IV.9), 0 -> R^r -> R^q -> R^{pX} ⊕ J R^t -> 0, with R/J gives an exact sequence 0 -> F2^t --δ--> F2^r -> F2^q -> F2^{pX+2t} -> 0, where δ is exactly the extension-class representative under the identification Ext^1(R^{pX}⊕JR^t,R^r) ≅ Hom_F2(F2^t,F2^r). Exactness forces δ to be injective, hence rank(δ)=t. This supplies the missing step in Lemma 6 and confirms both θ and eθ have rank t. A minor imprecision is that the remark on lifting F2 matrices by determinant-units is not literally true in a Laurent polynomial ring, but surjectivity of GL_n(R)→GL_n(F2) holds via lifting elementary matrix factorizations, so the conclusion is unaffected. Thus the central existence theorem does not have a identified correctness gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a symmetry-preserving decoupling theorem for 2D translation-invariant topological CSS codes on qubits. After passing to the maximal anyon-preserving superlattice, every code satisfying exactness, injectivity, finiteness of the anyon spaces, and the annihilator condition is shown to be chain-isomorphic to a standard complex consisting of t toric-code sectors together with p_Z Z-basis and p_X X-basis product-state sectors. The authors provide an explicit algorithm for computing the chain isomorphism in polynomial time in the coarse-grained parameters, prove bounds on supercell size and operator spreading, and demonstrate the construction on color codes and bivariate bicycle codes. Corollaries cover ancilla-free and shift-free circuit compilation, reabsorption of product states, periodic boundary conditions, and a decoder based on the decoupling map.","tokens_in":39182,"tokens_out":26473,"duration_ms":223001,"significance":"If the main theorem is correct, it closes a question left open by earlier classification results of Bombin, Haah, and others: after the anyon-permuting translations are removed by coarse-graining, no further obstruction to local unitary decoupling into toric codes and product states exists. The proof strategy is attractive: reduce modulo J, reinterpret the resulting short exact sequences as extensions, use the classification of extensions to prove existence, and then repair an explicit chain map to make it invertible. The paper gives reproducible numerical evidence and code, benchmarks on color codes and BB codes, and honest statements about empirical scaling. The central gap I identify is a missing step in the proof of Lemma 6; it is load-bearing but local and, in my view, fixable within the manuscript's scope.","major_comments":[{"comment":"The full-rank claim for θ is asserted but not proved. The text classifies the reference extensions E_Δ and asserts that pushouts and pullbacks 'traverse all matrices of the same rank', but it never shows that the representative of an arbitrary input extension has rank t; this is load-bearing because the transitivity of GL(r,F2) on full-rank r×t matrices is exactly what allows the pushout by φ2 to reach the standard extension. Please add the missing argument: applying the long exact sequence for Tor(-,F2) to the exact sequence (IV.9) yields 0 -> F2^t --δ--> F2^r -> F2^q -> F2^{pX+2t} -> 0, where δ is the connecting homomorphism and is injective by exactness; under the identification Ext^1(R^{pX}⊕JR^t,R^r) ≅ Hom_F2(F2^t,F2^r), δ is the extension-class representative, so rank(δ)=t. The same argument applies to eθ. With this step added, the proof of Lemma 6 is complete.","section":"Supplemental Sec. IV.C, Lemma 6"},{"comment":"As written, the sequence (IV.9) has right-hand term R^{pX}⊕JR^t, but the map ξ0^{-1}H_Xξ1 maps into R^s, not into R^{pX}⊕JR^t; identifying the image of this map with R^{pX}⊕JR^t is part of what Lemma 6 proves, so Lemma 5 cannot already assert this as a well-defined exact sequence. Please reformulate Lemma 5 with right-hand term im(ξ0^{-1}H_Xξ1) and state the isomorphism to R^{pX}⊕JR^t only after Lemma 6, or define the map to R^{pX}⊕JR^t explicitly after choosing the isomorphism supplied by Lemma 6.","section":"Supplemental Sec. IV.B, Eq. (IV.9)"}],"minor_comments":[{"comment":"The sentence explaining that the natural outputs are the inverses of (ψ2,ψ1,ψ0) is important, but the physical-interpretation paragraph immediately afterward refers to ψ1 as though it were the forward map; please make the convention explicit in both places.","section":"Main text, before Algorithm"},{"comment":"The phrase 'no additional ancillas in generic cases' is vague; Section VI.A gives precise sufficient conditions, so the abstract should carry the same qualification.","section":"Abstract and Section VI.A"},{"comment":"The string 'J144,12,12Kgross code' appears to be a typesetting error; it should refer to the gross code of Ref. [6].","section":"Main text, periodic-boundary paragraph"},{"comment":"The empirical exponents T∝q^1.86 and deg(ψ1^{-1})∝q^0.65 are fitted to a sample of 60 instances without stated confidence intervals; please state the sample size and, if possible, error bars, and make clear that these exponents are not used in the proof of the main theorem.","section":"Section V.C and Fig. SM.5"},{"comment":"The worst-case exponents d0^192 q^195 make the polynomial-time statement formally correct but practically uninformative; a remark stating that these bounds are not observed in the benchmarks would help readers calibrate the claim.","section":"Supplemental Sec. VII, Eq. (VII.4)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a strong and likely correct contribution; the central theorem is supported by a coherent proof whose only substantive gap is the compressed full-rank argument in Lemma 6, which I verified and which the authors should add. The independent concurrent work [51] is disclosed transparently, and I see no novelty or citation concerns. I recommend major revision rather than rejection because the gap is local and fixable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe bottom line: this paper proves what it claims. It closes the open question whether anyon permutation is the only obstruction to symmetry-preserving decoupling of 2D translation-invariant topological CSS codes. The result sharpens the known stack-of-toric-codes equivalence: after passing to the maximal anyon-preserving superlattice, the code is local-unitarily equivalent to toric codes plus product states, and the decoupling map can be computed in polynomial time. That is a genuine advance over Bombin and Haah's coarse-graining constructions, which break more translation symmetry than needed.\n\nWhat's new and good: the theorem itself, the explicit algorithm (Algorithm SM.1), the supercell bounds, and the worked examples on color codes and BB codes. The supplement is serious: reduction mod J, extension classification via Ext^1, chain-homotopy correction, Five Lemma. The proof is not just sketched; it is complete enough to be checked. The disclosure of concurrent independent work by Song is handled well.\n\nSoft spots, in proportion:\n\n- Lemma 6 is compressed. The claim that extension representatives have full rank t is load-bearing, and the proof in the supplement skips a step. The stress-test note fills it by tensoring the short exact sequence with R/J and using exactness to show injectivity of the extension representative. I agree with that; the lemma is correct, but a referee should ask the authors to expand the argument. The remark about lifting F2 matrices by determinant-units is not literally true in Laurent polynomial rings, but the needed surjectivity GL_n(R) -> GL_n(F2) holds via elementary matrices, so the conclusion stands. Minor.\n\n- The worst-case complexity exponents are enormous: time O(d^{192} q^{195}), degree O(d^{16} q^{16}). The authors are honest that these are conservative and that empirical scaling is far milder (q^1.86, q^0.65). Still, \"polynomial\" is formally true but practically meaningless for the worst case. This is a caveat, not a flaw.\n\n- The GitHub repository with code and data is referenced but not verified. The benchmark section is illustrative, not a rigorous empirical claim; they don't overstate it.\n\n- The decoder application is preliminary; they show it's faster but worse than BP-OSD, which is fine and honestly reported.\n\nThe citation pattern looks solid: Haah, Bombin, Kitaev, etc. are there, and the concurrent work is disclosed. No red flags.\n\nWho this is for: anyone working on classification of topological stabilizer codes, QCA, or decoding of 2D codes. It deserves a serious referee; the proof should be checked carefully, especially Lemma 6, but the central argument holds.\n\nMy recommendation: send to peer review. I'd encourage the editor to find a referee comfortable with homological algebra over Laurent polynomial rings, and to ask for a detailed proof of Lemma 6.\n\nBest,","headline":"A complete, constructive answer to the decoupling question at maximal anyon-preserving symmetry; the main theorem holds up, with minor proof-compression and complexity caveats.","tokens_in":39726,"tokens_out":2018,"would_cite":true,"duration_ms":17788,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P70","13P10","18G15"],"pacs":["03.67.Pp"],"model":"deepseek-v4-flash","headline":"This paper proves that after coarse-graining to the maximal anyon-preserving superlattice, every 2D translation-invariant topological CSS code is local-unitarily equivalent to a stack of toric codes and product states, and that the…","keywords":["translation-invariant topological CSS codes","toric code decoupling","anyon-preserving superlattice","local unitary equivalence","extension classification","Laurent polynomial ring","bivariate bicycle codes","Clifford quantum cellular automata"],"falsifier":"Run the paper's reduction modulo $J$ on any 2D translation-invariant topological CSS code that satisfies exactness, injectivity, finite anyons, and the annihilator condition, then compute the representative of its extension class in $\\mathrm{Hom}_{\\mathbb{F}_2}(\\mathbb{F}_2^t,\\mathbb{F}_2^r)$; if any representative has rank less than $t$, the claimed chain isomorphism to the standard toric-code stack cannot exist.","tokens_in":38769,"feed_emoji":"🧩","tokens_out":8442,"duration_ms":70221,"temperature":0.7,"pith_summary":"The paper sets out to prove that anyon-permuting translations are the only obstruction to splitting a 2D translation-invariant topological CSS code into simple building blocks. It shows that after coarse-graining to the largest superlattice whose translations leave every anyon type fixed, every such code is local-unitarily equivalent to a stack of toric codes together with product states. The proof is constructive: the authors give a polynomial-time algorithm that outputs the decoupling map, with explicit bounds on the supercell size and on how far the map spreads local Pauli operators. If true, this closes a gap left by earlier classifications, which achieved the same decoupling only after breaking more translation symmetry than necessary, and it gives a concrete route to decoding and logical-gate protocols for families such as bivariate bicycle codes.","feed_headline":"Toric-code split costs no extra translation symmetry","feed_subtitle":"After the maximal anyon-preserving coarse-graining, every 2D topological CSS code decouples locally; the algorithm is polynomial-time.","key_machinery":"The argument runs through the module-theoretic chain complex representation of CSS codes, where $R=\\mathbb{F}_2[x^{\\pm 1},y^{\\pm 1}]$ encodes the two translation directions. The load-bearing objects are reduction modulo the ideal $J=(x-1,y-1)$, which turns the topological data into finite-dimensional $\\mathbb{F}_2$ linear algebra; the classification of short exact sequences by $\\mathrm{Ext}^1_R(R^{p_X}\\oplus J R^t, R^r) \\cong \\mathrm{Hom}_{\\mathbb{F}_2}(\\mathbb{F}_2^t,\\mathbb{F}_2^r)$, whose representatives are $r\\times t$ matrices over $\\mathbb{F}_2$; and the transitivity of $\\mathrm{GL}(r,\\mathbb{F}_2)$ on full-rank matrices, which lets a pushout by an invertible change of stabilizer basis move the input extension to the standard extension. A chain-homotopy correction step then restores invertibility of the constructed maps, and the Five Lemma guarantees the middle map is an isomorphism.","core_discovery":"The central claim is that for any code whose chain complex and dual over $R=\\mathbb{F}_2[x^{\\pm 1},y^{\\pm 1}]$ are exact, whose check maps are injective, whose anyon spaces have common finite dimension $t$, and whose anyon annihilator is exactly the ideal $(x-1,y-1)$, there is a commuting chain isomorphism from the input complex to a standard complex built from $t$ toric-code sectors, $r-t$ $Z$-basis product states, and $s-t$ $X$-basis product states. The isomorphism is an $R$-module isomorphism, hence a translation-invariant local Clifford unitary; its inverses are computed by the paper's Algorithm SM.1 in time polynomial in $q$ and the input matrix degrees. Because the annihilator condition is precisely what holds on the maximal anyon-preserving superlattice, the theorem says the only symmetry breaking ever needed is the removal of anyon-permuting translations.","pith_inferences":["Beyond the paper: if the full-rank lemma is robust, the extension-class rank serves as a natural homological count of the genuine toric-code sectors, suggesting that $t$ can be identified from the extension class before any decoupling map is constructed.","Beyond the paper: the two-dimensionality of the result appears essential, since an independent concurrent classification noted that a naive three-dimensional analogue fails; symmetry-preserving decoupling may therefore be a genuinely planar phenomenon.","Beyond the paper: the decoder described here suggests a testable family of flushed decoders that first project syndromes onto $\\mathrm{coker}\\, H_X$; comparing their logical error rates with symmetry-based decoders would quantify how much accuracy the independent-sector approximation costs."],"forward_implications":["Anyon permutation is the sole obstruction: after passing to the maximal anyon-preserving superlattice, no further coarse-graining is needed to decouple a 2D translation-invariant topological CSS code into toric codes and product states.","The decoupling unitary can be compiled to a constant-depth circuit of CNOT gates with no ancillas in generic cases, and on a finite torus it maps the code to $t$ toric codes with $2t$ logical qubits.","Syndromes of the original code can be decoded by mapping them to independent toric-code sectors and running minimum-weight perfect matching, with a speed-accuracy tradeoff because local faults spread across sectors.","Logical-gate protocols, such as transversal Clifford gates, can be transported between different 2D topological CSS codes by composing them with the decoupling maps, at bounded loss of locality.","The algorithm's empirical running time and operator spreading on bivariate bicycle codes scale as $q^{1.86}$ and $q^{0.65}$, respectively."],"supporting_citations":[{"why":"Supplies the free-module chain-complex representation of translation-invariant CSS codes, including the injectivity and finite-anyon facts that the main theorem assumes.","marker":"[14]"},{"why":"Establishes the coarse-graining and anyon-hopping picture for 2D topological stabilizer codes, setting the background of previous decoupling results.","marker":"[9]"},{"why":"Provides a previous constructive classification that achieves decoupling by iterative coarse-graining, the extra symmetry breaking this paper removes.","marker":"[10]"},{"why":"Gives an earlier classification of translation-invariant topological Pauli stabilizer codes whose approach the theorem sharpens to the maximal residual symmetry.","marker":"[11]"},{"why":"Presents the unfolding of color codes into toric codes with ancillas, a motivating example whose ancilla requirement the generic construction removes.","marker":"[7]"},{"why":"Introduces bivariate bicycle code families used as benchmark instances in the numerical tests.","marker":"[6]"},{"why":"Provides degree bounds for Gröbner bases of modules, used to show the algorithm runs in polynomial time with controlled operator spreading.","marker":"[25]"},{"why":"Supplies the stability theorem used to decompose the constructed map into elementary matrices, i.e., constant-depth CNOT layers.","marker":"[30]"},{"why":"Gives an algorithmic proof of the same stability result, used for compiling the unitary into a circuit.","marker":"[31]"}],"fun_headline_variants":["No extra symmetry needed to split topological codes","Maximal anyon lattice unlocks local decoupling","Polynomial-time untangling for all 2D CSS codes","Translation-invariant split of CSS codes into toric blocks","Anyon-preserving superlattice removes all obstructions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on Lemma 6, which asserts that every admissible code's extension class is represented by an $r\\times t$ matrix of full rank $t$; should some code satisfying the theorem's conditions yield a lower-rank representative, the decoupling map would not exist.","fun_headline_variants_meta":{"raw":{"variants":["No extra symmetry needed to split topological codes","Maximal anyon lattice unlocks local decoupling","Polynomial-time untangling for all 2D CSS codes","Translation-invariant split of CSS codes into toric blocks","Anyon-preserving superlattice removes all obstructions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000256,"raw_usage":{"total_tokens":1535,"prompt_tokens":863,"completion_tokens":672,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":479,"completion_tokens_details":{"reasoning_tokens":596}},"tokens_in":479,"tokens_out":672,"duration_ms":5948,"temperature":1.0,"reasoning_tokens":596,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:22:52.406488+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the paper's reduction modulo $J$ on any 2D translation-invariant topological CSS code that satisfies exactness, injectivity, finite anyons, and the annihilator condition, then compute the representative of its extension class in $\\mathrm{Hom}_{\\mathbb{F}_2}(\\mathbb{F}_2^t,\\mathbb{F}_2^r)$; if any representative has rank less than $t$, the claimed chain isomorphism to the standard toric-code stack cannot exist.","supporting_citations":[{"cited_title":"Therefore the superlattice is generated byx 2,xy","cited_arxiv_id":null,"evidence_quote":"Supplies the free-module chain-complex representation of translation-invariant CSS codes, including the injectivity and finite-anyon facts that the main theorem assumes."},{"cited_title":"Thus the superlattice is generated byx 3,x 2y","cited_arxiv_id":null,"evidence_quote":"Establishes the coarse-graining and anyon-hopping picture for 2D topological stabilizer codes, setting the background of previous decoupling results."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides a previous constructive classification that achieves decoupling by iterative coarse-graining, the extra symmetry breaking this paper removes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives an earlier classification of translation-invariant topological Pauli stabilizer codes whose approach the theorem sharpens to the maximal residual symmetry."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Presents the unfolding of color codes into toric codes with ancillas, a motivating example whose ancilla requirement the generic construction removes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces bivariate bicycle code families used as benchmark instances in the numerical tests."}],"review_version":2}