{"id":"da1ed4e6-bdda-4796-9a13-037a93faf076","arxiv_id":"2610.03677","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper gives an explicit, deterministic recovery protocol for Kitaev non-Abelian surface codes for any finite group, conditional on pre-identified neutral error clusters and perfect syndrome measurements.","lead":"This paper constructs explicit quantum circuits that detect and correct errors in non-Abelian surface codes, a class of quantum error correction codes built from the quantum double of any finite group. It fills a gap: earlier work proved such correction is possible but did not say how to do it.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"Theorem 6 and the Sec. 3.3 protocol rest on an unformalized structural lemma for arbitrary cluster topologies; no error found in substance, but a small-lattice verification would settle the universality claim.","rationale":"I read the full text, including Appendices A and B. Prop. 1 (single-edge basis), Prop. 3 (matrix-unit vertex algebra), the gauge-fixing family (2.46) with completeness (Lemma 5), and the KL theorem (Thm. 6, (2.48)) are carefully argued and self-consistent. The transfer circuits check out algebraically: Lemma 9 verifies the conjugation identities (3.10), the flow (3.8)–(3.15) and (3.27)–(3.34) are consistent when re-derived, and the Z2 and S3 examples confirm the cases i→v and i←v and all four plaquette edge positions. The protocol handles superposed cluster errors by linearity: each basis element maps to |label⟩⊗|ψ⟩, so the data sector is restored regardless of the ancilla outcome. Gauge-fixing, irreps, and the Peter-Weyl Fourier transform carry no solvability requirement, supporting the non-Abelian/A5 claim. The acknowledged scoping — decoder and syndrome input (b26, b54), linear cluster-size scaling (b342), no threshold or measurement-error analysis (b344) — is honestly stated and does not contradict the internal claim. The remaining risk is the unformalized structural lemma in Thm. 6 and the general-protocol induction for arbitrary cluster topologies, which is the place a hidden counterexample would live; the forest-leaf structure makes the argument sound in substance, and branched geometries also work by correcting around branch points. Because the substantive argument holds up and the identified concern is a verification gap rather than a detected error, the reader's ACCEPT stands UNCHANGED; I rank this internal structural point above the decoder assumption as the primary correctness risk precisely because the decoder is explicitly scoped out, whereas the structural lemma is load-bearing for the universal claim.","tokens_in":34343,"tokens_out":14164,"duration_ms":85708,"concrete_test":"On a small patch (e.g. 5×5 with opposite rough/smooth boundaries), enumerate all gauge-fixed completed error supports ⃗i containing no logical operator, focusing on topologies beyond Fig. 1: a dual tree with a degree-3 or degree-4 branch point, a dual cycle with a tail, and two disconnected components. For G = S3 (and optionally A5), compute P_c(⃗i)(E^†F)P_c(⃗i) for all gauge-fixed basis elements and verify (2.48) holds as δ·P_c(⃗i); then simulate the Sec. 3.3 flux-then-charge transfer circuits (3.14), (3.34) acting on E|ψ⟩ for random code states |ψ⟩ and verify the data qudits are returned to |ψ⟩ with the ancillas carrying the error labels. If any enumerated topology violates (2.48) or fails to return the data to the code space, the universality claim is restricted to the demonstrated geometries; if all pass, the structural lemma and the induction are verified for these topologies.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim — exact, deterministic recovery for any finite group G and any neutral cluster — is carried internally by Theorem 6 (the gauge-fixed KL condition (2.48)) plus the transfer circuits (3.14), (3.34) and the sequential protocol of Sec. 3.3. The load-bearing internal assumption is the structural step in the proof of Theorem 6 (§2.5): for any gauge-fixed completed support ⃗i with no dual/direct loop and no logical operator, there is always a check whose support contains exactly one errored edge (\"there is a plaquette p ∈ p(⃗i) that contains exactly one of these errored edges\", and analogously a vertex for the diagonal part). This lemma is argued verbally via Figs. 1–2, and the Sec. 3.3 \"General Protocol\" then asserts — also by analogy with the ≤2-star/≤2-plaquette demonstrations — that the sequential inward correction terminates correctly for any cluster topology, invoking exactly that single-errored-edge check at each round. The argument is sound in substance: the multiplication-error edge set is a forest in the dual graph whose only excluded leaf-pair is a path between two smooth-boundary plaquettes (which would be a logical operator), so a usable leaf exists, and edge removal preserves the invariants; branched trees also work because branch-point plaquettes are never required to be single-error checks. But the universality claim rests entirely on this unformalized lemma and induction, so a hidden counterexample would live here. The decoder input is stated as an assumption (b26, b54) and is out of scope; perfect syndrome and the absence of measurement-error analysis are acknowledged (b344). The primary residual correctness risk is therefore the formalization gap of the structural lemma and of the general-protocol induction, which the paper only demonstrates on the specific configurations of Fig. 1 and the two combined cases.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper studies exact recovery for Kitaev quantum double D(G) surface codes for arbitrary finite group G. It constructs an orthogonal error basis on each G-qudit using left-multiplication operators L_h and rescaled irrep operators E_µ (Eq. (2.11)), and introduces a gauge-fixing on multi-edge supports that removes stabilizer redundancies, yielding the family E_i of Eq. (2.46). Theorem 6 states that this gauge-fixed family satisfies the Knill-Laflamme conditions with identity matrix on any completed support that does not host logical operators (Eq. (2.48)). Section 3 constructs charge and flux transfer circuits (Eqs. (3.14) and (3.34)) that move error labels onto ancilla qudits, and a sequential protocol that corrects neutral clusters by applying these transfers starting from boundary checks. Appendix B carries the same construction over to ribbon errors. The paper explicitly assumes as input a decoder that supplies neutral, correctable error clusters and perfect syndrome measurements (Sec. 1, sentences b45/b54), and it does not construct that decoder.","tokens_in":34882,"tokens_out":7586,"duration_ms":85613,"significance":"If correct, this is a substantial advance: it would provide the first explicit, deterministic recovery protocol for non-Abelian quantum double surface codes for arbitrary finite groups, including non-solvable groups such as A5, where previous work established existence of recovery or treated specific groups. The algebraic core is careful and largely self-contained: the single-edge orthogonality (Proposition 1), the matrix-unit algebra (Proposition 3), the charge and flux commutation identities (Lemmas 2 and 4, Eqs. (3.6), (3.28)), and the Knill-Laflamme verification for the constructed error families are concrete and checkable. The paper also deserves credit for being explicit that the decoder input is an external assumption and for discussing the linear-scaling limitation of the sequential protocol in the Conclusion.","major_comments":[{"comment":"The proof of Theorem 6 (Eq. (2.48)) rests on an unformalized structural assertion: for any completed support i with no dual/direct loop and no logical operator, there is always a plaquette p∈p(i) whose boundary contains exactly one errored edge, and similarly a vertex whose star contains exactly one diagonal errored edge. This is argued verbally through Figs. 1–2 but not proved. Because the same assertion is reused in §3.3 to justify termination of the General Protocol, the paper's central universality claim for arbitrary neutral clusters depends on this lemma. A formal graph-theoretic proof, or a systematic verification over all small supports, is required before the claim can be considered established.","section":"§2.5, Theorem 6 proof","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The General Protocol states that sequential inward correction terminates and returns the code space for any gauge-fixed support, citing the single-error-check property at each round. The induction over cluster topology is not written out: one must show that after each flux or charge transfer, the remaining error set still contains no dual/direct loop and no logical operator, and that the chosen order of transfers preserves these invariants. The worked examples (one star, two stars, one plaquette, two plaquettes, separated and combined configurations) are not a proof of the general case. This is load-bearing for the 'exact and deterministic recovery protocol' claim and should be proved or explicitly verified.","section":"§3.3 General Protocol","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The hypotheses of Theorem 6 are stated as 'any completed support i that does not host any logical operator', but the notion of 'hosts a logical operator' and the single-error-check assertion are not made precise near boundaries, where checks may be trivial (K=G smooth or K={id} rough) and Eq. (2.36) changes the counting. The proof of Theorem 6 does not explicitly handle boundary paths ending on smooth or rough boundaries. Since the patches described in §2.1 have such boundaries, the theorem's hypotheses and proof need to be made precise for boundary supports.","section":"§2.5 and §2.4 boundary hypotheses","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The heading 'identifies' should be 'identities'.","section":"§A.3","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"'Rbbon operators' should be 'Ribbon operators'.","section":"Fig. 5 caption","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"'te support' should be 'the support'.","section":"§B.1","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"In the two-plaquette flux example, the text reads 'we we can correct all multiplication errors'; the duplicated 'we' should be removed.","section":"§3.3","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The condition 'a≤n_K^Γ or b≤n_K^Γ equal δ_{a,b}' is ambiguous; it should state that the matrix elements equal δ_{a,b} when both indices lie in the K-invariant subspace, and vanish when exactly one does.","section":"Eq. (2.36)","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The main open point is the unformalized structural lemma behind Theorem 6 and the General Protocol. If the authors provide a rigorous proof or a small-support verification, I would support acceptance; the algebraic and circuit derivations appear sound. The paper's abstract and introduction state the protocol as applicable to any neutral cluster, which is slightly stronger than what is proved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"This is a construction paper, not an existence argument, and that is the point. Prior work (Cui et al. [27]) proved recovery exists for every finite G; Kim and McGinley [28] explicitly left construction open. This paper fills that gap: an explicit orthogonal error basis for D(G) surface codes with gauge-fixing to kill stabilizer redundancy (Lemma 5), Knill-Laflamme verified directly (Theorem 6), and charge/flux transfer circuits (3.14, 3.34) that move every free label onto an ancilla. The S3 example is the most reassuring part—it shows the ancilla-data entanglement before controlled multiplication, which is exactly where a naive derivation breaks, and they handle it correctly. The ribbon extension in App. B is substantive, not decorative. The limitations are stated prominently: decoder and syndrome extraction are assumed inputs (b26, b54), correction is linear in cluster size (b342), thresholds deferred (b344). No fitted parameters, no circularity, no invented entities. Citation pattern is clean; self-citations are background use of their own prior lattice-surgery work.\n\nOne real soft spot. The proof of Theorem 6 and the general protocol in Sec. 3.3 both rest on an unformalized structural claim: any completed support with no dual/direct loop and no logical operator contains a check whose support has exactly one errored edge. The \"Hence\" in the Theorem 6 proof is where the argument lives, and it is carried by Fig. 1 and verbal reasoning. The Sec. 3.3 protocol is demonstrated on the specific configurations, then asserted for arbitrary cluster topology. I have stared at it and the substance is sound—the multiplication-error edge set is a forest in the dual graph, the only excluded leaf-pair is a path between smooth-boundary plaquettes (which would be a logical), and branched trees work because branch points are never needed as single-error checks. But sound-in-substance is not proved-in-text, and the universality claim is carried entirely by this gap. A small-lattice brute-force verification, even just for one non-solvable group, would settle it. That is a referee request, not a rejection reason.\n\nThe paper does not overclaim. The abstract, intro, and conclusion all state the decoder is an assumed input; the exact deterministic recovery is for clusters already identified, and the full fault-tolerant pipeline is explicitly out of scope.\n\nThis is for anyone working on QEC for topological codes, particularly the recent non-Abelian surface code push for constant-depth non-Clifford gates. The machinery here will be used. Send it out; ask the referee to scrutinize Theorem 6's structural step and request the small-lattice check.","headline":"A real construction paper: explicit error basis and transfer circuits for D(G) surface codes for any finite group, with one genuine but non-fatal gap—the universality argument in Theorem 6 rests on an unformalized structural lemma.","tokens_in":36273,"tokens_out":937,"would_cite":true,"duration_ms":29158,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["81P70","81P68","20C15"],"pacs":["03.67.Pp"],"model":"deepseek/deepseek-v4-flash","headline":"This paper claims exact, deterministic error recovery for the quantum double surface code $D(G)$ of any finite group $G$, including non-solvable groups like $A_5$, via a gauge-fixed orthogonal error basis and ancilla transfer circuits.","keywords":["non-Abelian surface codes","quantum double","error basis","gauge fixing","Knill-Laflamme conditions","exact recovery","ancilla transfer circuits","finite group representations"],"falsifier":"A direct test is to compute the local Knill-Laflamme matrix (2.48) on a small patch: if any off-diagonal entry is non-zero for a completed support with no direct or dual loop and no logical operator, Theorem 6 fails. A cheaper combinatorial check is to search small lattices for a completed support where every plaquette contains two or more errored edges or every vertex star does, which would break the proof's single-errored-edge step. The full experimental check is to run the circuits (3.14) and (3.34) on a $D(S_3)$ or $D(A_5)$ cluster under ideal measurements and verify that the data qudits return exactly to the code state and the ancilla outcomes equal the applied labels $(\\vec{h}, \\vec{\\mu})$ with certainty.","tokens_in":35930,"feed_emoji":"⚛️","tokens_out":13907,"duration_ms":135206,"temperature":0.7,"pith_summary":"This paper claims that error correction for the quantum double surface code $D(G)$ can be made exact and deterministic for every finite group $G$, including non-Abelian and non-solvable groups such as $A_5$. Such codes can host non-Clifford transversal gates, so exact recovery is the missing piece for fault-tolerant universal computation with them; earlier work proved that a recovery exists or found thresholds, but did not construct it. The paper removes the obstruction, which is that non-invertible anyon fusion means syndromes seeing only anyon occupation miss internal labels, by constructing a complete, orthogonal, gauge-fixed error basis on the code space and proving that it satisfies the Knill-Laflamme conditions. It then supplies charge and flux transfer circuits that move every error label onto its own ancilla, returning the data qudits exactly to the code space and reading out the full labels. The authors acknowledge that their protocol assumes a neutral, correctable error cluster and perfect syndrome measurements as input, with the cluster-identifying decoder left to future work; the recovery layer itself, however, works uniformly for any finite group.","feed_headline":"Exact error correction built for every finite-group surface code","feed_subtitle":"Now even non-Abelian groups like A5 get exact recovery, not just existence proofs.","key_machinery":"The load-bearing object is the gauge-fixed error family $E_{\\vec{i}}$ of Eq. (2.46): on each vertex star one edge is fixed to carry no multiplication error ($h = \\mathrm{id}$) and on each plaquette one edge is fixed to carry no irrep error ($\\mu = 1$), after which the family has exactly $|G|^{2N - |\\vec{i}_A| - |\\vec{i}_B|}$ elements and is complete and orthogonal on the code space. The orthogonality proof of Theorem 6 rests on a graph fact: a completed support with no direct or dual loop and no logical operator always contains a plaquette whose boundary meets exactly one errored edge and a vertex whose star meets exactly one, so leftover products are annihilated by a single check before any identity term survives. The circuits that carry the recovery are the charge transfer $T = F_{v_a} CL_{i,v_a} U_v$, where $U_v$ is the controlled vertex operator and $F$ the non-Abelian Fourier transform, and the flux transfer $CL^\\dagger_{p_a,i} V_p$, where $V_p$ is the controlled plaquette holonomy; both are built entirely from controlled left and right multiplications plus one Fourier transform, and they move irrep (charge) and multiplication (flux) labels onto vertex and plaquette ancillas respectively.","core_discovery":"For a surface code based on the quantum double $D(G)$ of a finite group $G$, the paper constructs exact recovery in the strict Knill-Laflamme sense. The natural error basis on each edge, $E^{h,\\mu}_i = L^h_i E^{\\mu}_i$ with $E^{\\mu}_i = \\sqrt{d_\\Gamma} Z^{\\Gamma}_{ab,i}$ (left multiplication combined with a rescaled irrep matrix element), is overcomplete on the code space: stabilizers create redundancies whenever errors close into loops on the lattice or dual lattice. The paper's gauge-fixing picks one fixed edge per vertex star, whose multiplication label is set to the identity, and one per plaquette, whose irrep label is set to the trivial irrep, and completes every error support so that no direct or dual loop remains; Lemma 5 shows every error is equivalent up to stabilizers to this gauge-fixed form. Theorem 6 then proves that the Knill-Laflamme matrix of the gauge-fixed family is the identity on the code space, so errors on a neutral cluster are mutually orthogonal and exactly correctable. The recovery circuits make this explicit: a charge transfer circuit $T = F_{v_a} CL_{i,v_a} U_v$ (with orientation-dependent conjugation) moves an irrep label onto a vertex ancilla, and a flux transfer circuit $CL^\\dagger_{p_a,i} V_p$ moves a multiplication label onto a plaquette ancilla, so processing a cluster from its boundary inward places each of the $2N - |\\vec{i}_A| - |\\vec{i}_B|$ free labels on its own ancilla and returns the data to the code space; the same two-stage protocol corrects ribbon errors with two ancillas. The paper states explicitly that the input is a neutral, correctable cluster identified by a decoder acting under perfect syndrome measurements, a component it leaves to future work.","pith_inferences":["A natural next test, which the paper does not run, is to couple these transfer circuits to an unoptimized cluster-finder on $D(S_3)$ and measure whether the round-by-round shrinking of clusters converts the linear-in-cluster-size cost into a threshold; the paper explicitly leaves this numerical verification open.","Because the transfer circuits use only controlled multiplications and one Fourier transform, they may compose with the adaptive constant-depth charge-measurement circuits already known for solvable groups, suggesting a fully gate-efficient syndrome-plus-recovery pipeline for solvable groups.","The gauge-fixing analysis is phrased for trivial 2-cocycles, but the group law enters only through the flatness identity (2.45), so a plausible extension is exact recovery for twisted quantum doubles with non-trivial cocycles such as $D(\\mathbb{Z}_3^2, \\omega_{III})$ using the same basis and transfer idea.","The per-label ancilla accounting saturates the error-space dimension, hinting that the protocol is locally optimal in ancilla usage; one could test this by proving that any exact recovery circuit for a completed cluster requires at least $2N - |\\vec{i}_A| - |\\vec{i}_B|$ fresh qudits when the error labels are unknown."],"forward_implications":["For any finite group $G$, including non-solvable ones like $A_5$, errors supported on a neutral cluster with completed support and no logical operator can be corrected exactly and deterministically, in the sense that the code state is returned intact and the error labels are read out on ancillas.","Correction consumes exactly $2N - |\\vec{i}_A| - |\\vec{i}_B|$ ancillas per cluster, one per free error label, matching the dimension (2.47) of the gauge-fixed error space, so no error label is measured twice and none is left unmeasured.","Ribbon (anyon-string) errors, which carry a single label pair on an extended one-dimensional support, are corrected by the same flux-then-charge protocol using only two ancillas, and the protocol also covers ribbons ending on rough and smooth boundaries.","Because correction moves inward from the cluster boundary one layer per round, the authors note that the circuits can be applied iteratively to shrink large clusters by a constant amount per round, a mode the paper suggests could yield a threshold.","The protocol applies to boundary errors for any boundary subgroup $K$, where the irrep labels collapse from $|G|$ to $|K|$ on the code space, so the same recovery works on patches with mixed rough and smooth boundaries.","pith_inferences are editorially distinguished from paper claims in the following bullets."],"supporting_citations":[{"why":"Supplies the quantum-double surface-code model itself, its vertex and plaquette Hamiltonians, and the ribbon algebra that the error basis and correction circuits are built from.","marker":"[1]"},{"why":"Provides the quantum error-correction conditions restated as Theorem 7, whose satisfaction is the paper's exact-recovery criterion.","marker":"[49]"},{"why":"Establishes that local errors are correctable for every finite group and that code distance grows with system size, the topological background that guarantees correctability of short errors.","marker":"[27]"},{"why":"Supplies the boundary condensations by subgroups $K$ and the ribbon commutation relations at boundary sites used in Secs. 2.4 and B for boundary and ribbon errors.","marker":"[44]"},{"why":"Sets up the square-patch boundary conditions encoding a logical group algebra $\\mathbb{C}[G]$, the specific code geometry the error basis must respect.","marker":"[21]"},{"why":"Shows why syndromes seeing only anyon occupation miss internal degrees of freedom, the specific gap the full-label error basis fills.","marker":"[38]"},{"why":"Provides non-Abelian surface codes realizing non-Clifford transversal gates, the application context that makes exact QEC for general groups necessary.","marker":"[24]"},{"why":"Proves recovery thresholds exist for non-Abelian codes without constructing the recovery map, the existence-versus-construction gap this paper closes.","marker":"[28]"}],"fun_headline_variants":["Exact recovery proven for every finite-group surface code","Non-Abelian surface codes now have exact, deterministic correction","Gauge-fixing unlocks exact recovery for quantum double codes","Knill-Laflamme exactness achieved for all finite-group surface codes","Charge and flux circuits give exact recovery for non-Abelian codes"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The claim collapses if the unbuilt decoder fails to supply neutral, correctable clusters under the assumed perfect syndrome measurements, because the transfer circuits only act on clusters already identified as such; the proof of Theorem 6 also relies on a verbally-argued graph fact that a loop-free, logical-operator-free completed support always has a check meeting exactly one errored edge.","fun_headline_variants_meta":{"raw":{"variants":["Exact recovery proven for every finite-group surface code","Non-Abelian surface codes now have exact, deterministic correction","Gauge-fixing unlocks exact recovery for quantum double codes","Knill-Laflamme exactness achieved for all finite-group surface codes","Charge and flux circuits give exact recovery for non-Abelian codes"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000184,"raw_usage":{"cost":0.000184408,"is_byok":false,"cost_details":{"upstream_inference_cost":0.000184408,"upstream_inference_prompt_cost":0.000121408,"upstream_inference_completions_cost":6.3e-05},"total_tokens":1297,"prompt_tokens":1072,"completion_tokens":225,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":256,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":137}},"tokens_in":1072,"tokens_out":225,"cost_source":"openrouter_usage","duration_ms":2344,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":137,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-05T04:37:44.070867+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"A direct test is to compute the local Knill-Laflamme matrix (2.48) on a small patch: if any off-diagonal entry is non-zero for a completed support with no direct or dual loop and no logical operator, Theorem 6 fails. A cheaper combinatorial check is to search small lattices for a completed support where every plaquette contains two or more errored edges or every vertex star does, which would break the proof's single-errored-edge step. The full experimental check is to run the circuits (3.14) and (3.34) on a $D(S_3)$ or $D(A_5)$ cluster under ideal measurements and verify that the data qudits return exactly to the code state and the ancilla outcomes equal the applied labels $(\\vec{h}, \\vec{\\mu})$ with certainty.","supporting_citations":[{"cited_title":"Fault-tolerant quantum computation by anyons","cited_arxiv_id":"quant-ph/9707021","evidence_quote":"Supplies the quantum-double surface-code model itself, its vertex and plaquette Hamiltonians, and the ribbon algebra that the error basis and correction circuits are built from."},{"cited_title":"A Theory of Quantum Error-Correcting Codes","cited_arxiv_id":"quant-ph/9604034","evidence_quote":"Provides the quantum error-correction conditions restated as Theorem 7, whose satisfaction is the paper's exact-recovery criterion."},{"cited_title":"Kitaev's quantum double model as an error correcting code","cited_arxiv_id":"1908.02829","evidence_quote":"Establishes that local errors are correctable for every finite group and that code distance grows with system size, the topological background that guarantees correctability of short errors."},{"cited_title":"The Quantum Double Model with Boundary: Condensations and Symmetries","cited_arxiv_id":"1006.5479","evidence_quote":"Supplies the boundary condensations by subgroups $K$ and the ribbon commutation relations at boundary sites used in Secs. 2.4 and B for boundary and ribbon errors."},{"cited_title":"Hybrid Lattice Surgery: Non-Clifford Gates via Non-Abelian Surface Codes","cited_arxiv_id":"2510.20890","evidence_quote":"Sets up the square-patch boundary conditions encoding a logical group algebra $\\mathbb{C}[G]$, the specific code geometry the error basis must respect."},{"cited_title":"Intrinsic Heralding and Optimal Decoders for Non-Abelian Topological Order","cited_arxiv_id":"2507.23765","evidence_quote":"Shows why syndromes seeing only anyon occupation miss internal degrees of freedom, the specific gap the full-label error basis fills."},{"cited_title":"Constant-Depth Clifford-Hierarchy Gates via Non-Abelian Surface Codes","cited_arxiv_id":"2512.13777","evidence_quote":"Provides non-Abelian surface codes realizing non-Clifford transversal gates, the application context that makes exact QEC for general groups necessary."},{"cited_title":"Mixed-state topological order and error-correction thresholds in non-Abelian codes: rigorous results","cited_arxiv_id":"2607.21706","evidence_quote":"Proves recovery thresholds exist for non-Abelian codes without constructing the recovery map, the existence-versus-construction gap this paper closes."}],"review_version":1}