{"id":"b8f90c91-bd3b-4570-9f87-3b601119e1cc","arxiv_id":"2505.13599","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new logical-observable matching decoder lets surface codes run fast transversal Clifford gates while correcting all errors below half the code distance, and windowed variants trade efficiency against reset speed.","lead":"Quantum error correction usually forces a slowdown between logical gates; this paper shows a surface-code decoder that works across fast transversal Clifford gates. The authors run minimum-weight matching separately for each logical measurement and benchmark it on random Clifford circuits.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The d/2 guarantee for the lom decoder rests on an unproven decoding-subgraph property (Appendix E); if G_O ever contains a projected hyperedge, MWPM cannot be applied and Theorem 1 collapses.","rationale":"The reader's weakest assumption identifies the same load-bearing gap: the unproven claim that G_O is a graph. Theorem 1's proof is otherwise standard, and the numerical benchmarks are extensive, but they cannot substitute for the missing structural verification because a single 3-vertex projection would break the MWPM reduction. The windowed-decoder issues are explicitly labeled conjectural in the paper, so they are not the basis of the central claim; the central d/2 claim is the lom decoder, and its soundness rests on the G_O property and the informal fragile-observable reduction. I therefore agree with the reader's conditional verdict and do not see a reason to strengthen or weaken it. The concrete enumeration test is cheap, decisive for the tested gate set, and directly addresses the weakest link.","tokens_in":52456,"tokens_out":22444,"duration_ms":232327,"concrete_test":"Use the released Stim-based code to enumerate the full basic decoding hypergraph in the pre-gate frame for a battery of circuits: each single gate G in {I, H, S, CNOT}, plus random depth-3 two-qubit Clifford compilations, for d=5 and d=7. For every logical observable O in the independent reliable generating set, including X- and Z-basis observables and products, compute V_O and test that |h∩V_O| is in {0,2} for every hyperedge h. If any hyperedge projects to 1 or 3 vertices, the graph property is false; if none does, the property is supported but still needs a formal inductive proof over detector frames and gate types.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1 assumes that for every reliable observable O, the decoding subgraph G_O is a graph. The main text calls this 'straightforward... by visual inspection' and Appendix E gives only an intuitive parity argument. The gap is that the parity argument does not rigorously enumerate all hyperedge types and all observable types. CNOT hyperedges connect vertices on different logical qubits, and observables whose backpropagated Pauli is Y-like (e.g. X-basis observables passing through S gates) have H_O containing both X- and Z-type boundary edges, so V_O contains both detector types. The argument needs an explicit correspondence between h∩V_O and the overlap of the associated boundary space-time stabilizer with H_O; without it, a weight-3 hyperedge could project to 3 vertices, in which case E_O is a hypergraph and single-lom is not a matching instance. I found no explicit counterexample, but the load-bearing property is not settled. If it fails, the splitting failure discussed in Section IVC shows the logical error rate can degrade to O(p), so the central d/2 claim would collapse for that circuit.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a minimum-weight-perfect-matching (MWPM) based decoder, called logical observable matching (lom), for decoding the unrotated surface code when transversal Clifford gates are applied with a single QEC round between gates. The central construction is to project the full decoding hypergraph onto a subgraph G_O associated with each reliable logical observable O, then run independent MWPM instances on these subgraphs. The paper claims that, under a basic independent X/Z error model, the lom decoder corrects any error of weight less than d/2 for arbitrary circuits built from fast transversal Clifford gates and T-gate injections. It also introduces windowed variants, analyzes failures of hierarchical and splitting-hyperedge decoders, and presents extensive numerical benchmarks under phenomenological and circuit-level depolarizing noise, including comparisons with minimum-weight decoding.","tokens_in":52630,"tokens_out":9674,"duration_ms":101184,"significance":"If the central claim holds, this is an important step toward practical decoding of fast transversal logic: it replaces minimum-weight hypergraph decoding by standard MWPM on carefully chosen subgraphs, and it provides a concrete route to decoding with one QEC round per transversal gate. The paper is strong in several respects: the proofs of Lemma 1 and Theorem 1 are clearly structured, the numerical study is extensive with confidence intervals, thresholds, and comparisons to minimum-weight decoding, and the authors are transparent about the conjectural status of the windowed decoder variants and about the circuit-level distance-reducing errors for the repeated-S experiment. The main weakness is that the load-bearing structural property that each projected subgraph G_O is a graph is not rigorously proved; the current Appendix E gives an intuitive parity argument rather than a formal proof. This gap, together with the informal treatment of fragile observables in full circuits, means the d/2 guarantee is conditional in a way that should be resolved before the central claim is accepted as proven.","major_comments":[{"comment":"The claim that E_O = {h ∩ V_O : h ∈ H, h ∩ V_O ≠ ∅} is a set of edges for every reliable observable O and every circuit built from {H,S,CNOT} with one gate per round is load-bearing for Theorem 1, but it is not proven. The main text calls this 'straightforward... by visual inspection' and refers to the Supplemental Material, yet Appendix E explicitly presents only an 'intuitive, general, argument.' The argument is not logically sufficient: the fact that a space-time stabilizer has even overlap with the observing region H_O does not imply that each individual weight-3 hyperedge h satisfies |h ∩ V_O| even, since two hyperedges with odd overlap could cancel in the total stabilizer. Please provide a rigorous proof, e.g. by enumerating all hyperedge types in the basic error model (weight-1, weight-2, and weight-3 time-like hyperedges from I, H, S, and CNOT in the pre-gate frame) and verifying for each observable type that |h ∩ V_O| ≤ 2. Until this is supplied, the statement that the single-lom decoder runs MWPM on a matching instance, and hence the d/2 guarantee, is conditional.","section":""},{"comment":"Theorem 1 is stated and proved only for a single reliable observable. The extension to an arbitrary full circuit, including fragile observables, conditioning measurements, and T-gate injections, is described algorithmically but not formalized. In particular, the procedure of randomly assigning an outcome to a fragile observable, then decoding a reliable product observable and inferring the original outcome, requires a proof that no basic error of weight < d/2 can flip the final corrected logical outcome. The discussion of replacing a fragile-conditioned S gate by a Pauli gate tracked in software is an argument sketch rather than a lemma. Please state and prove a theorem for the full lom decoder on arbitrary circuits, or explicitly mark this extension as a conjecture. This matters because the Introduction's central claim is about 'arbitrary circuits' and not only about a single reliable observable.","section":""}],"minor_comments":[{"comment":"The text says the numerical computation of |e_min| verifies that the decoder is 'circuit-distance preserving,' but then notes that for the repeated-S experiment in the X-basis under circuit-level noise, |e_min| = 2,4 for d = 3,5, i.e. d-1. This is acknowledged, but it would be clearer to state in the Introduction or Abstract that the d/2 guarantee applies to the basic error model and that circuit-level noise can reduce the effective distance by one in specific cases.","section":""},{"comment":"The caption states that the decoded Z-measurement observable 'happens to be fragile,' which may confuse readers because Section III.B2 says fragile observables need not be decoded. Consider choosing a reliable observable in the figure or explaining explicitly why the figure is pedagogical despite the observable being fragile.","section":""},{"comment":"The year in reference [63] appears as '20245'; this should be '2025'.","section":""},{"comment":"The efficiency argument for the basic windowed-lom decoder uses the function f(t) quantifying operator spreading, but f(t) is only defined later in Appendix B2. Define f(t) at first use in Section V.B1 or add a forward reference.","section":""},{"comment":"The discussion notes that more than one S and/or CNOT gate per QEC round can produce weight-4 hyperedges in G_O, which prevents reduction to matching. This is an important scope limitation of the lom decoder and should be mentioned prominently, perhaps in the abstract or introduction, so that readers do not assume the decoder works for arbitrary gate densities.","section":""}],"recommendation":"major_revision","confidential_remarks":"This is a valuable and mostly well-executed manuscript. The central construction is promising, and the numerical study is extensive and carefully reported. However, the graph property of G_O is genuinely load-bearing for Theorem 1 and is not proven at the required level of rigor; the current Appendix E is an intuitive sketch. I do not see evidence that the claim is false, and the issue may well be fixable with a finite enumeration or a more careful stabilizer-overlap argument. Likewise, the full-circuit treatment of fragile observables should be formalized. For these reasons I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The LOM decoder is the real thing: a matching-based decoder that runs separate MWPM instances for each reliable logical observable and simply declines to decode fragile ones. That gets you a d/2 fault-tolerance guarantee for arbitrary transversal Clifford circuits with one QEC round per gate, which no previous matching decoder achieved. The proof of Theorem 1 is clean given its assumptions, and the numerical work is extensive and careful—thresholds close to memory experiments, explicit comparisons to minimum-weight decoding, and public code and data. The section on why hierarchical matching decoders fail via time-like loops is a useful contribution in its own right.\n\nThe soft spots are real but mostly disclosed. The load-bearing claim that G_O is always a graph is argued by visual inspection and a parity argument in Appendix E, not proved. I read the argument and I do not see an actual counterexample, but the property is exactly where I would expect a gap to hide: the parity argument does not fully enumerate hyperedge types intersecting V_O. If a projected hyperedge ever survives, single-lom stops being a matching instance and Theorem 1 collapses for that observable. That is a load-bearing unproven lemma, and I would want it proved before building anything on it.\n\nThe windowed decoders are presented honestly as partial. The basic windowed decoder is efficient only with slow resets, the two-step version gives up efficiency, and the d/2 guarantee for both is Conjecture 1 supported by short-cut edges but not proven. The time-like snake constructions are real and interesting, but they also show the windowed route is not finished. These limitations are stated in the paper clearly, which I respect.\n\nMinor: the repeated-S X-basis experiment loses one distance unit to a hook error, and circuit-level noise can reduce the effective distance by a constant factor. Both are disclosed and quantified.\n\nWho is this for? Anyone working on decoding transversal gates, fast logic, or surface-code implementations on neutral atoms or trapped ions. The core LOM result deserves refereeing: it is new, it works as far as I can check, and the proof strategy is transparent enough to verify. I would not desk-reject it, but I would send it to a referee who will push hard on Appendix E and on Conjecture 1. If those hold up, this becomes a standard citation for fast logical decoding.","headline":"A genuinely new per-observable matching decoder for fast transversal Clifford gates, with an honest but incomplete treatment of the one property the main theorem depends on.","tokens_in":53158,"tokens_out":1455,"would_cite":true,"duration_ms":18128,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows that decoding across fast transversal Clifford gates in the unrotated surface code can be done with minimum-weight perfect matching on per-observable subgraphs, preserving the code-distance guarantee of d/2.","keywords":["surface code","transversal Clifford gates","logical observable matching","minimum-weight perfect matching","windowed decoding","fragile observables","decoding hypergraph","fast logical gates"],"falsifier":"Search exhaustively over small unrotated surface codes and circuits with two H, S, or CNOT gates between QEC rounds for a reliable observable O whose projected subgraph G_O contains a weight-4 hyperedge under the basic error model; if one exists, the graph assumption fails and the matching-based d/2 guarantee collapses in that setting.","tokens_in":52200,"feed_emoji":"⚛️","tokens_out":5228,"duration_ms":55005,"temperature":0.7,"pith_summary":"The paper shows that a surface-code decoder can keep up with fast transversal Clifford gates and T-gate injections without losing the code's distance, even though the raw decoding problem contains hyperedges that ordinary matching cannot handle. Its logical observable matching (lom) decoder runs a separate minimum-weight perfect matching instance for each reliable logical measurement, on a matchable subgraph built by projecting the decoding hypergraph onto the region where errors can flip that measurement. For circuits made of H, S, and CNOT gates with one QEC round per gate, the paper proves that any basic error of weight less than d/2 is corrected. Numerical benchmarks under phenomenological and circuit-level depolarizing noise show thresholds close to memory experiments, for both repeated and arbitrary two-qubit Clifford circuits. Windowed versions of the decoder are also proposed, with trade-offs between computational efficiency, reset speed, and fault-tolerance that are only conjecturally resolved.","feed_headline":"Fast Clifford gates decoded at full surface-code distance","feed_subtitle":"Decoding each logical observable separately corrects every error below half the code distance, with one QEC round per gate.","key_machinery":"The load-bearing object is the decoding subgraph G_O for an observable O, defined by taking the observing edge set H_O — the basic errors that flip O, which lie on one spatial boundary — and including all detectors of the same Pauli type at the same logical circuit locations, then projecting every hyperedge of the full decoding hypergraph onto those vertices. For one gate per QEC round in the {H,S,CNOT} set, this projection contains only edges, so minimum-weight perfect matching applies. The proof of Theorem 1 then follows the familiar surface-code distance argument: if the combined error-and-correction string had odd overlap with H_O, it would have to connect the two spatial boundaries and hence contain at least d edges, which contradicts the bound w(error) < d/2 together with minimality of the correction. The pre-gate detector frame keeps detecting regions local in space-time, which is what makes the observing edge set a small, boundary-local set of edges.","core_discovery":"The central claim is that decoding across arbitrary sequences of fast transversal Clifford gates in the unrotated surface code can be reduced to independent minimum-weight matching problems, one per logical observable, without sacrificing the d/2 correction radius. For any reliable observable, the single-lom decoder projects the full decoding hypergraph onto a subgraph G_O that contains the observing edge set of that observable and only same-time, same-Pauli-type detectors; for the {H,S,CNOT} gate set with one gate per QEC round this projection is a graph, not a hypergraph. Theorem 1 then proves that the decoder correctly predicts whether an error of weight below d/2 flips the observable, using the standard surface-code argument that a mistaken logical flip would require a correction path of weight at least d. Fragile observables are never decoded directly: their outcomes are sampled randomly and combined with later observables so that only reliable products are decoded. The paper also shows that naive hyperedge-splitting and hierarchical matching decoders have distanceless failure patterns, and that windowed variants need additional short-cut edges and synchronized resets or measurements to avoid sublinear-weight logical failures.","pith_inferences":["The matchable-subgraph projection is a general decoding technique: since the paper notes it works with any graph-based decoder, it likely extends to Union-Find and to rotated or color codes with appropriate boundary structures.","The 'independent observers' framing suggests that any windowed decoder whose windows overlap in space-time can suffer from time-like snakes; decoders with non-overlapping commit regions avoid this failure mode by construction.","A testable extension is to benchmark the windowed-lom decoder with short-cut edges under circuit-level noise and compare thresholds to the non-windowed lom decoder, which would show whether the conjectured fault-tolerance is practically relevant.","The sublinear-weight failure examples imply that distance alone is not a reliable proxy for windowed-decoder performance; circuit-by-circuit validation may be necessary for fast-logic decoding strategies."],"forward_implications":["The lom decoder sustains full code distance while running one QEC round per transversal gate, so fast logical gates do not force a Θ(d) slow-down for decoding.","Naive splitting of weight-3 hyperedges and hierarchical matching decoders are not fault-tolerant in this setting; the lom decoder avoids their constant-weight and distance-independent logical failure patterns.","Numerically, thresholds under phenomenological and circuit-level depolarizing noise are close to those of memory experiments for repeated and random two-qubit Clifford circuits, suggesting that transversal gates need not degrade logical performance.","The basic windowed-lom decoder is computationally efficient under slow resets, while the two-step variant handles fast resets but may be inefficient; both require synchronization and short-cut edges to conjecturally correct all errors of weight below d/2."],"supporting_citations":[{"why":"Proves that O(1) QEC rounds between transversal gates and magic-state injections can be fault-tolerant, setting the target regime for the lom decoder.","marker":"[10]"},{"why":"Introduces correlated decoding of logical algorithms with transversal gates, the earlier approach that the lom decoder makes matching-based.","marker":"[9]"},{"why":"Describes splitting decoders for hypergraph faults, the method the paper shows is not fault-tolerant for transversal-gate circuits.","marker":"[23]"},{"why":"The stabilizer circuit simulator used to generate detector error models, hyperedge decompositions, and all numerical syndrome data.","marker":"[24]"},{"why":"A concurrent windowed decoder for fast transversal gates that uses heuristic approximate hypergraph solvers rather than matching, providing the comparison point for windowed-lom.","marker":"[11]"},{"why":"The original sliding-window matching decoder for memory experiments, whose window structure the windowed-lom decoder adapts.","marker":"[12]"}],"fun_headline_variants":["MWPM per logical observable decodes Clifford gates at full distance","Full correction radius for fast transversal Clifford gates","Decode any Clifford gate sequence without losing distance","One MWPM per observable: fast gates, full d/2 radius"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The d/2 guarantee relies on the decoding subgraph G_O for every observable O being a true graph with no hyperedges for circuits of {H,S,CNOT} gates with one gate per QEC round, a property the paper supports by inspection and an intuitive argument rather than a full proof.","fun_headline_variants_meta":{"raw":{"variants":["MWPM per logical observable decodes Clifford gates at full distance","Full correction radius for fast transversal Clifford gates","Decode any Clifford gate sequence without losing distance","One MWPM per observable: fast gates, full d/2 radius"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000249,"raw_usage":{"total_tokens":1564,"prompt_tokens":972,"completion_tokens":592,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":588,"completion_tokens_details":{"reasoning_tokens":526}},"tokens_in":588,"tokens_out":592,"duration_ms":6688,"temperature":1.0,"reasoning_tokens":526,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:13:22.533542+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search exhaustively over small unrotated surface codes and circuits with two H, S, or CNOT gates between QEC rounds for a reliable observable O whose projected subgraph G_O contains a weight-4 hyperedge under the basic error model; if one exists, the graph assumption fails and the matching-based d/2 guarantee collapses in that setting.","supporting_citations":[{"cited_title":"descending staircase","cited_arxiv_id":null,"evidence_quote":"Proves that O(1) QEC rounds between transversal gates and magic-state injections can be fault-tolerant, setting the target regime for the lom decoder."},{"cited_title":"descending staircase","cited_arxiv_id":null,"evidence_quote":"Introduces correlated decoding of logical algorithms with transversal gates, the earlier approach that the lom decoder makes matching-based."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Describes splitting decoders for hypergraph faults, the method the paper shows is not fault-tolerant for transversal-gate circuits."},{"cited_title":"1 ford= 3, with spatial coordinates(x,y)withx,y∈1 2 Z and0≤x,y≤d−1","cited_arxiv_id":null,"evidence_quote":"A concurrent windowed decoder for fast transversal gates that uses heuristic approximate hypergraph solvers rather than matching, providing the comparison point for windowed-lom."},{"cited_title":"For a memory experiment without logical gates, all frames are equivalent and the detectors are defined in the standard way, i.e","cited_arxiv_id":null,"evidence_quote":"The original sliding-window matching decoder for memory experiments, whose window structure the windowed-lom decoder adapts."}],"review_version":1}