{"id":"f466ed03-e07c-4520-93ff-c5f1f5a78291","arxiv_id":"2610.05901","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For the D4 quantum double under E charge noise, refining syndromes from anyon-location-only to full onsite charges to pair-fusion outcomes raises the decoding threshold from about 0.111 to 0.165 to 0.214, and enables single-shot full logical qudit recovery below the minimum-weight perfect…","lead":"This paper designs explicit maximum-likelihood error-correcting decoders for a non-Abelian topological code and shows that measuring richer syndrome information, including how non-Abelian anyons fuse, systematically raises the noise threshold at which the encoded qubit is protected.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"The single-shot full-qudit claim (Sec. III E, Eq. 117) rests on an unproven confinement assumption and a MWPM threshold borrowed from a different lattice model; a direct simulation of the single-shot protocol on this paper's D4 cylinder would settle whether p_full >= 0.103 actually holds.","rationale":"My independent read locates the same weak point the reader flagged: the single-shot full-qudit recovery claim (Sec. III E) is the least-secure load-bearing assertion because it is the one headline result that is argued, not computed, and it depends on (i) a confinement assumption the authors state but do not prove and (ii) an MWPM threshold 'borrowed' from Ref. [50] on a different lattice, which they themselves disclaim. I verified this against the full text: Sec. III E contains no independent numerical threshold for p_full, and the supporting reference is explicitly a different model. The central-qubit threshold hierarchy, by contrast, is backed by exact tensor-network contractions, a recovered Nishimori exponent for S(1), and careful finite-size analysis including parity-dependent corrections and Rao-Blackwellized estimators; I found no internal inconsistency there worth elevating. The fusion-disturbance result (Sec. III D, general-group statement that noncentral holonomy is detectable while the central qubit is preserved) is a clean derivation and is not in question. Because the full-qudit concern is honestly acknowledged rather than hidden, and because the numerical core stands, the appropriate verdict remains CONDITIONAL: the central claims are substantially supported, but the 'single-shot recovery of the full logical qudit at nonzero threshold' headline is not independently demonstrated for this model at the claimed value. My test is a concrete, in-model simulation using the paper's own machinery, which would settle whether p_full >= p_MWPM holds or whether Eq. (117) needs revision. I do not see grounds to move beyond CONDITIONAL in either direction.","tokens_in":64453,"tokens_out":6776,"duration_ms":64191,"concrete_test":"Directly measure the single-shot full-qudit failure probability on the exact model of this paper: the L x L D4 quantum double cylinder with smooth boundaries, E-charge noise, and the one-round syndrome P_~S = P_{f,HMWPM} P_{S(2)} (fusion outcomes along HMWPM paths measured once, as in Eq. 118). For p in 0.06-0.13 and L = 4-12: sample noise, record P_~S, apply the recovery, and test whether the full five-dimensional logical state is recovered. Extract p_full by finite-size scaling (the Appendix F machinery already fits three such transitions) and compare directly with p_MWPM ~ 0.103 rather than borrowing it. If the extrapolated p_full is statistically consistent with >= 0.10, the headline claim and confinement assumption are confirmed in-model; if p_full falls below ~0.08, Eq. (117) must be revised and the centered claim requires re-statement.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's second headline result, single-shot recovery of the full five-dimensional logical qudit below a nonzero threshold, is established only through an argument, not a computation. Concretely, Eq. (117) states p_full >= p_HMWPM >= p_MWPM ~ 0.103, and the authors explicitly say the MWPM value is borrowed: 'because the lattice model is different from our D4 quantum double, we cannot directly use their threshold' (Sec. III E). The argument then relies on the stated-but-unproven confinement assumption: below the HMWPM threshold, the union of physical error strings and correction paths decomposes into contractible, vacuum-fusing components, so 'the subsequent Abelian cleanup remains confined to these components' and 'cannot distinguish logical states or implement a nontrivial logical operation.' The only support cited is Ref. [50]'s observation of statistically consistent thresholds with and without Abelian correction 'in its D4 realization' — again a different lattice. The load-bearing risk is concrete: if confinement holds only up to a strictly smaller p, or if the fusion-first 'measure before transport' protocol (Eq. 119, P_~S = U-dagger Q-lambda U) contaminates coherences between noncentral logical sectors, then p_full < p_MWPM and the stated lower bound in Eq. (117) is literally false, leaving the headline full-qudit claim unsupported at the claimed value. This is an acknowledged-but-unresolved gap, not a claimed error: the central-qubit threshold hierarchy (Eq. 5, p_c = 0.111 / 0.165 / 0.214) is numerically resolved, independently benchmarked through the S(1) Nishimori exponent (nu = 1.548 vs 1.50(3)), and is not what I am questioning. I also note the Rényi-2 intrinsic-threshold discussion is explicitly flagged by the authors as not establishing the von Neumann threshold, so it does not carry the claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper develops maximum-likelihood (ML) decoders for non-Abelian topological order, working out the D4 quantum double on a cylinder under incoherent E-charge noise. It constructs three nested syndrome records — a coarse E/no-E onsite record S(1), a complete onsite irrep record S(2), and a record S(3) that additionally resolves pairwise E-anyon fusion outcomes — and derives their Born probabilities as classical partition functions (Sec. III, Appendices A–D). Numerical finite-size scaling gives central-qubit thresholds p_c^(1) ≈ 0.111 < p_c^(2) ≈ 0.165 < p_c^(3) ≈ 0.214 (Eq. 5), with a consistency check from the S(1) transition recovering the known Nishimori exponent (ν = 1.548 vs 1.50(3)). The paper further argues that a fusion-first 'measure before transport' protocol allows single-shot recovery of the full five-dimensional logical qudit below p_MWPM ≈ 0.103 (Eq. 117), and computes a Rényi-2 coherent-information proxy suggesting an intrinsic transition near p_c ≈ 0.46. The central-qubit threshold hierarchy is the well-supported core of the paper; the single-shot full-qudit claim is the load-bearing weak point.","tokens_in":64862,"tokens_out":7488,"duration_ms":64361,"significance":"If it stands, the central-qubit result is a genuine advance: it demonstrates concretely that resolving non-Abelian fusion outcomes systematically raises ML decoding thresholds, and it does so with a rigorous derivation of syndrome probabilities, exact tensor-network contractions (bond dimension 5), and a clean logical derivation of the threshold ordering from syndrome nesting (Eq. 132) rather than from fitted circularity. The reproduction of the Nishimori exponent within error bars is a strong independent check on the numerical pipeline. The threshold hierarchy itself is credible and falsifiable. The second headline claim — single-shot recovery of the full logical qudit below a nonzero threshold — is not yet established at the stated value: it rests on an unproven confinement assumption and a threshold value borrowed from Ref. [50] for a different lattice model. The paper is honest about both facts, but the abstract presents the claim as established.","major_comments":[{"comment":"The paper's second headline claim — single-shot recovery of the full five-dimensional logical qudit below a nonzero threshold — is established only by an argument, not by a computation on the model studied. Eq. (117) asserts p_full ≥ p_HMWPM ≥ p_MWPM ≃ 0.103, where the numerical value 0.103 is taken from Ref. [50] for a different lattice; the text states this directly: 'because the lattice model is different from our D4 quantum double, we cannot directly use their threshold.' The chain then rests on the stated-but-unproven confinement assumption that below the HMWPM threshold the union of physical error strings and correction paths decomposes into contractible, vacuum-fusing components, so that 'the subsequent Abelian cleanup remains confined to these components' and 'cannot distinguish logical states or implement a nontrivial logical operation.' The only cited support is again Ref. [50]'s observation of statistically consistent thresholds in its own D4 realization, which is a different lattice. If confinement holds only up to a strictly smaller p, or if the fusion-first 'measure before transport' projector of Eq. (119) (P_~S = U†QλU) disturbs coherences between the noncentral logical sectors, then p_full < p_MWPM and the stated lower bound in Eq. (117) is false. Sec. V A itself lists proving single-shot success even up to the S(2) threshold as open. A direct simulation of the single-shot protocol on the D4 cylinder, even at moderate sizes, is needed to substantiate the claim at the claimed value.","section":"Sec. III E, Eq. (117)","authors_acknowledge":"It would be interesting to study optimal recovery of the full logical qudit in more depth, and to try to prove the success of single-shot decoding at least up to theS (2) threshold (sinceS (2) would not in- clude such dangerous fusion measurements).","authors_acknowledge_p":0.79},{"comment":"The intermediate inequality p_HMWPM ≥ p_MWPM in the chain (117) is asserted, not established for the model studied. The text says only 'All we can say is that pHMWPM ≥ pMWPM ≃ 0.103 on the square lattice,' a statement made in Ref. [50]'s setting, not for the D4 cylinder of this paper. Because the single-round fusion-first protocol of Eq. (119) replaces the two-round HMWPM protocol's explicit coherent transport, the algebraic identity P_~S = U†QλU does not by itself guarantee that the single-round measurement reproduces the two-round performance for every syndrome and every logical input; the success of the protocol at p near 0.103 is an empirical question that the paper does not address numerically. This step of the chain is load-bearing for Eq. (117) and needs either a proof in this model or direct numerical evidence.","section":"Sec. III E, inequality p_HMWPM ≥ p_MWPM","authors_acknowledge":"It would be interesting to study optimal recovery of the full logical qudit in more depth, and to try to prove the success of single-shot decoding at least up to theS (2) threshold (sinceS (2) would not in- clude such dangerous fusion measurements).","authors_acknowledge_p":0.7}],"minor_comments":[{"comment":"The abstract and the 'Main results' list state the full-qudit single-shot claim unconditionally ('We show that there is a nonzero noise threshold below which the entire logical qudit can be recovered with a single round of measurements'), without the caveats that Sec. III E attaches to it: the unproven confinement assumption and a threshold borrowed from a different lattice model. The wording should be aligned with the hedged discussion in Sec. III E and V A.","section":"Abstract and Sec. I A","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"Figure 1 presents the intrinsic threshold at p_c ≃ 0.46 as a schematic 'information-theoretic threshold,' but Sec. IV E explicitly states that the Rényi-2 diagnostic weights the spectrum differently from the von Neumann coherent information and that the apparent coincidence 'does not establish equality of the intrinsic recovery thresholds.' The figure and caption should carry a clearer caveat that this scale is a Rényi-2 proxy, not an established intrinsic threshold.","section":"Fig. 1 and Sec. IV E","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The open question in Sec. V A — proving single-shot success at least up to the S(2) threshold — is in tension with the abstract's unconditional claim of single-shot recovery of the full qudit. The authors should reconcile the two statements, for example by explicitly characterizing the Sec. III E full-qudit result as conjectural at the stated threshold value.","section":"Sec. V A","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The acknowledgements note coordination with Ref. [71]; novelty overlap with that work should be checked editorially. The central-qubit threshold hierarchy is the solid core of the paper and is well supported by the analytic derivations, the exact contractions, and the Nishimori-exponent consistency check. The single-shot full-qudit claim is the risky part: it rests on an unproven confinement assumption plus a threshold borrowed from a different lattice model, and Eq. (117) as a numerical statement is not supported in this model. I would recommend requiring either a direct numerical demonstration of the single-shot protocol on the D4 cylinder or an explicit demotion of that claim to a conjecture before acceptance. The abstract currently overclaims relative to the body, and this should be fixed in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"You should know two things about arXiv:2610.05901. It is the first explicit maximum-likelihood decoder with refined syndromes for a non-Abelian quantum double, and the central-threshold results hold up. The other headline — single-shot recovery of the full five-dimensional logical qudit — is an argument, not a demonstrated computation, and the authors say so themselves.\n\nWhat's actually new: three nested syndrome records for the D4 quantum double under E charge noise — S(1) records E/non-E per site, S(2) the complete onsite charge, S(3) adds pair-fusion outcomes — with numerically resolved thresholds 0.111, 0.165, 0.214. The ordering is a strict logical consequence of nested information (Eq. 132), not a fit. The S(1) transition independently recovers the known Nishimori exponent (nu = 1.548 vs 1.50(3)), a strong consistency check on the numerics. The most genuinely new physics is Sec. III D: fusion measurements along homologically distinct paths can read out noncentral logical sectors even though the measurement paths do not wind around the cylinder. The general-group statement (Eqs. 106 and 113) is clean, and the central-qudit decoding via Z_N charge grading (Sec. III F, with a D6 example) makes the framework extend beyond D4.\n\nThe soft spots, in proportion. Sec. III E's claim p_full >= p_MWPM ~ 0.103 rests on an unproven confinement assumption: after E recovery, Abelian cleanup supposedly decomposes into contractible vacuum-fusing components that cannot implement a logical operation. The MWPM threshold itself is borrowed from Ref. [50], which uses a different lattice, and the paper states it cannot directly use that value. The stress-test note is right: a direct simulation of the single-shot protocol on this paper's own D4 cylinder would settle it, and no such simulation is presented. This is an acknowledged gap, not a hidden error, and it does not touch the central-qubit hierarchy. Minor items: the S(3) large-size numerics use antithetic paired sampling, the finite-size fits carry model-choice uncertainty (the reported intervals are conditional on the ansatz), and the Rényi-2 intrinsic-threshold discussion is explicitly not a von Neumann threshold statement. The authors flagged both themselves.\n\nWho gets value: anyone working on decoders for non-Abelian topological order or on mixed-state phases of quantum doubles. The general-group derivations in Appendices A–D are checkable and the paper ships the numerical protocol. The reader's CONDITIONAL verdict is the right call.\n\nRecommendation: send it to peer review. A referee should push Sec. III E toward a direct simulation or a proof of the confinement assumption; the rest is in good shape. I would take it to reading group.","headline":"First explicit refined-syndrome ML decoders for a non-Abelian quantum double: the central-qubit threshold hierarchy is solid and well-benchmarked, but the single-shot full-qudit claim is an acknowledged argument a direct simulation should settle.","tokens_in":65748,"tokens_out":4588,"would_cite":true,"duration_ms":53155,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["81P70","81P68","82B20"],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"Refined syndrome records that resolve the fusion outcomes of non-Abelian anyon pairs systematically raise maximum-likelihood decoding thresholds, reaching about 0.214 for the central qubit of the D4 quantum double under E charge noise on…","keywords":["non-Abelian topological order","quantum double model","maximum-likelihood decoding","anyon fusion measurement","decoding threshold","charge decoherence","single-shot decoding","central logical qudit"],"falsifier":"Compute the von Neumann coherent information of the decohered $D_4$ state at larger system sizes than the Rényi-2 proxy allows and check whether the full-qudit and central-qubit intrinsic thresholds coincide near $p \\approx 0.46$, or instead separate; alternatively, add an independent incoherent Abelian charge noise channel and test whether the three syndrome thresholds converge toward one another as the authors conjecture.","tokens_in":65810,"feed_emoji":"⚛️","tokens_out":5430,"duration_ms":41169,"temperature":0.7,"pith_summary":"Maximum-likelihood error correction is harder for codes with non-Abelian anyons than for the toric code, because the locations of local excitations do not exhaust the error information and the logical operators cannot be classified by Abelian homology. This paper argues that the fix is to measure more: refining the syndrome record to include the fusion outcomes of non-Abelian anyon pairs systematically raises the decoding threshold. Concretely, for the central qubit of the $D_4$ quantum double under two-dimensional $E$-charge noise on a cylinder, three nested syndrome records give thresholds $p_c^{(1)} \\approx 0.111 < p_c^{(2)} \\approx 0.165 < p_c^{(3)} \\approx 0.214$. Below the minimum-weight perfect matching threshold, the full five-dimensional logical qudit can be recovered in a single round of measurements. The paper also establishes a structural cost: fusion measurements of incorrectly paired anyons can disturb logical information outside the central qubit, so the high threshold is bought by sacrificing the rest of the qudit.","feed_headline":"Adding fusion measurements lifts a non-Abelian code's threshold to 0.214","feed_subtitle":"Three refined syndrome records give the D4 central qubit thresholds of 0.111, 0.165, and 0.214.","key_machinery":"The central object is the pair-fusion charge projector $P^{\\lambda}_{p_{i,j}}$, built from ribbon operators $F_{p_{i,j}}(h)$ that leave the edges along a chosen path untouched while acting on the path's endpoints with parallel-transported group elements; it generalizes the onsite charge projector and measures the total electric charge of a strip containing a pair of $E$ anyons, thereby resolving the fusion channel in $E \\times E = A_{00} \\oplus A_{10} \\oplus A_{01} \\oplus A_{11}$. The technical workhorse is the reduction of every syndrome probability to a classical partition function — an Ising-type model for $S^{(1)}$ and a decorated loop model with closed-component neutrality constraints for $S^{(2)}$ and $S^{(3)}$ — and the evaluation of homology-resolved likelihoods $Z_S(h)$ via twisted partition functions that insert a $\\mathbb{Z}_2$ sign along a seam crossing the cylinder. The same machinery yields a Rényi-2 coherent-information diagnostic that locates the intrinsic information-theoretic transition near $p_c \\approx 0.46$ for both the full qudit and the central qubit.","core_discovery":"The paper claims that refined syndrome sets, including measurements that resolve non-Abelian anyon fusion outcomes, systematically improve the performance of maximum-likelihood decoders for non-Abelian topological order. In the $D_4$ quantum double decohered by the non-Abelian $E$ charge on a cylinder, the three nested records — recording only where the $E$ anyons sit ($S^{(1)}$), resolving the complete onsite irrep charge including the four Abelian anyons ($S^{(2)}$), and further measuring pairwise fusion outcomes of the $E$ anyons ($S^{(3)}$) — give progressively higher thresholds $p_c^{(1)} \\approx 0.111$, $p_c^{(2)} \\approx 0.165$, $p_c^{(3)} \\approx 0.214$. A further claim is that the full five-dimensional logical qudit is single-shot recoverable below the minimal-weight perfect matching threshold, by collapsing the two rounds of an earlier protocol — $E$-anyon annihilation followed by Abelian cleanup — into a single round of up-front fusion measurements. The paper also establishes a trade-off: because two homologically distinct fusion paths can differ by a noncontractible loop, a windings-loop fusion measurement acts as a projective measurement on the noncentral logical sectors while leaving the central qubit untouched, so protecting the central qubit at high threshold necessarily sacrifices the rest of the qudit.","pith_inferences":["The refined syndromes carry a non-local measurement cost: $S^{(3)}$ pairs arbitrarily distant $E$ anyons, so the threshold gain from 0.165 to 0.214 is bought with measurement weight that may be impractical in hardware; a natural extension is to estimate how much of the gain survives with only finite-range fusion measurement.","If the authors' conjecture is correct, adding independent incoherent Abelian charge noise should drive the three syndrome thresholds together until they coincide, which would make fusion-assisted decoding effective specifically for noise biased toward non-Abelian charges — a directly testable prediction.","The sacrificial-subspace structure suggests a general design principle for non-Abelian decoders: guarantee one protected logical sector at a high threshold and let the fate of the other sectors be decided by which anyons the measurement protocol happens to pair.","The Rényi-2 coincidence at $p \\approx 0.46$ for the full qudit and the central qubit could be an artifact of the Rényi-2 weighting of the spectrum, so the genuinely distinguishing calculation — a von Neumann coherent-information evaluation at larger sizes — remains open."],"forward_implications":["For any non-Abelian topological order whose non-Abelian anyons fuse only to Abelian outcomes, such as quantum doubles of nilpotent groups of class 2 including $D_4$, fusion measurements make single-shot full-logical-space recovery realistic below the MWPM threshold.","The central-qudit recipe — protecting the subspace associated with the group center via $\\mathbb{Z}_N$ homology while cleaning up other anyons arbitrarily — applies beyond nilpotent groups, as shown for $D_6$, where an $E_1$ error network carries a $\\mathbb{Z}_2$ winding variable even though $E_1 \\times E_1$ includes a non-Abelian outcome.","The threshold hierarchy $p_c^{(1)} \\approx 0.111 < p_c^{(2)} \\approx 0.165 < p_c^{(3)} \\approx 0.214$ shows that Abelian charges in the $S^{(2)}$ record act as hard constraints on where non-Abelian error strings can run, a mechanism that works because the Abelian anyons appear only at junctions of $E$ strings.","Above the MWPM threshold, fusion-based recovery necessarily sacrifices the noncentral three-dimensional part of the qudit, so decoder design becomes a deliberate choice of which logical subspace to protect.","The improved decoders still fall short of the intrinsic information-theoretic threshold near $p \\approx 0.46$, so the syndrome-refinement strategy is a way to approach, not saturate, the decodability limit of the topological order."],"supporting_citations":[{"why":"Defines the quantum double models based on finite groups $G$ that the paper specializes to the $D_4$ case throughout.","marker":"[3]"},{"why":"Supplies the maximum-likelihood decoding framework and the random-bond Ising model mapping for the toric code that the paper generalizes to non-Abelian order.","marker":"[4]"},{"why":"Provides the heralded MWPM protocol for the $D_4$ quantum double whose two-round correction the paper collapses into a single round of up-front fusion measurements, and whose threshold serves as the benchmark.","marker":"[50]"},{"why":"Sets the intrinsic information-theoretic threshold for decohered non-Abelian topological order that the paper argues syndrome-based decoders cannot saturate.","marker":"[55]"},{"why":"Supports the physical relevance of ground-state-encoded non-Abelian order and the torus anyon-creation-transport-fusion mechanism underlying the logical disturbance by fusion measurements.","marker":"[31]"},{"why":"Supplies the Nishimori-line random-bond Ising transition value used as a numerical benchmark to validate the $S^{(1)}$ threshold fitting.","marker":"[67]"}],"fun_headline_variants":["Fusion-aware decoding lifts D4 threshold to 0.214","Resolving fusion outcomes boosts non-Abelian decoder threshold","Single-shot recovery via fusion measurements below p=0.214","Non-Abelian anyon fusion data sharpens max-likelihood decoding"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The single-shot full-qudit claim rests on the stated but unproven confinement assumption that after the $E$-anyon recovery, the subsequent Abelian cleanup remains confined to contractible, vacuum-fusing components and therefore cannot distinguish logical states or implement a nontrivial logical operator.","fun_headline_variants_meta":{"raw":{"variants":["Fusion-aware decoding lifts D4 threshold to 0.214","Resolving fusion outcomes boosts non-Abelian decoder threshold","Single-shot recovery via fusion measurements below p=0.214","Non-Abelian anyon fusion data sharpens max-likelihood decoding"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.00019,"raw_usage":{"cost":0.000190008,"is_byok":false,"cost_details":{"upstream_inference_cost":0.000190008,"upstream_inference_prompt_cost":0.000121128,"upstream_inference_completions_cost":6.888e-05},"total_tokens":1316,"prompt_tokens":1070,"completion_tokens":246,"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":170}},"tokens_in":1070,"tokens_out":246,"cost_source":"openrouter_usage","duration_ms":3784,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":170,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-06T21:25:20.781011+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Compute the von Neumann coherent information of the decohered $D_4$ state at larger system sizes than the Rényi-2 proxy allows and check whether the full-qudit and central-qubit intrinsic thresholds coincide near $p \\approx 0.46$, or instead separate; alternatively, add an independent incoherent Abelian charge noise channel and test whether the three syndrome thresholds converge toward one another as the authors conjecture.","supporting_citations":[{"cited_title":"Intrinsic heralding and optimal decoders for non-abelian topolog- ical order, Physical Review Letters136, 120405","cited_arxiv_id":null,"evidence_quote":"Provides the heralded MWPM protocol for the $D_4$ quantum double whose two-round correction the paper collapses into a single round of up-front fusion measurements, and whose threshold serves as the benchmark."},{"cited_title":"Stability and loop models from decohering non-abelian topological order, Phys. Rev. Lett.134, 250403","cited_arxiv_id":null,"evidence_quote":"Sets the intrinsic information-theoretic threshold for decohered non-Abelian topological order that the paper argues syndrome-based decoders cannot saturate."},{"cited_title":"Non-abelian topologi- cal order and anyons on a trapped-ion processor, Nature 626, 505–511","cited_arxiv_id":null,"evidence_quote":"Supports the physical relevance of ground-state-encoded non-Abelian order and the torus anyon-creation-transport-fusion mechanism underlying the logical disturbance by fusion measurements."},{"cited_title":"Two-dimensional random- bond Ising model, free fermions, and the network model, Physical Review B65, 054425","cited_arxiv_id":null,"evidence_quote":"Supplies the Nishimori-line random-bond Ising transition value used as a numerical benchmark to validate the $S^{(1)}$ threshold fitting."}],"review_version":1}