{"id":"b2c6045b-dce7-41aa-83a1-b4e50948f606","arxiv_id":"2607.13784","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Any prime-dimensional qudit Pauli code can be foliated into a graph state for fault-tolerant measurement-based quantum computing.","lead":"This paper gives a recipe for turning any quantum error-correcting code made of high-dimensional quantum digits (qudits) into a graph state that can be processed by measurements. It matters because measurement-based qudit quantum computing is a promising route for photonic platforms, where high-dimensional states and measurements are natural.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Fault-tolerance claim is only established under perfect graph-state preparation plus post-preparation Z errors; photonic loss and fusion failures are outside the noise model.","rationale":"The reader's weakest assumption is exactly this noise-model gap, and I agree it is the most load-bearing limitation. The mathematical foliation construction itself appears coherent for the worked stabilizer and non-CSS examples, and the simplified-noise threshold simulations provide real evidence for the construction under that model. I do not see an internal inconsistency in Definitions 1 or 4 that would sink the framework; the universal proof for arbitrary subsystem and dynamical codes is not fully supplied, but that is a separate, addressable gap. The practical fault-tolerance claim, however, is tied to the photonic motivation and is not supported under realistic preparation and loss noise. The appropriate verdict therefore remains conditional, matching the reader's assessment.","tokens_in":18102,"tokens_out":15746,"duration_ms":151077,"concrete_test":"Using the authors' sdim simulator, simulate the foliated d=3 toric code with a preparation-level noise model: each CZ gate used to construct the graph state fails (or is followed by a depolarizing error) with probability p, or each qudit is erased with probability p_loss before the measurement stage. Recompute the logical error rate curves and threshold for lattice sizes D=8,10,12,14,16,18. If the threshold drops below the reported 0.023 for d=3, or if no threshold is visible, the perfect-preparation assumption is load-bearing and the fault-tolerance claim for photonic MBQC is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract claims that the foliated graph state 'can be measured to perform fault-tolerant measurement-based quantum computing', with photonics as the motivating platform. The only fault-tolerance evidence is in Appendix A: the graph state Λ is prepared perfectly, then Pauli-Z errors are applied to each qudit with probability p, and Proposition 3 maps these errors to circuit-level data and measurement errors. This noise model excludes the dominant photonic error mechanisms: photon loss, failed probabilistic fusion, and errors during the CZ entangling operations used to build the graph state. A lost or unentangled qudit is not equivalent to a post-preparation Z error, so the detector analysis in Eqs. (2) and (S.1) does not cover it. The conclusion itself notes that a photonic implementation would require joining smaller pieces via destructive fusion, but no fault-tolerance analysis of fusion failures or loss is provided. Thus the step from 'detectors work under simplified Z noise' to 'fault-tolerant MBQC for photonic qudits' is not supported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a framework for 'foliating' arbitrary Pauli-based quantum error-correcting codes over prime-dimensional qudits, extending earlier qubit constructions by Brown and Roberts. In the CSS case (Definition 1) it constructs a weighted qudit graph state whose measurements implement the code checks, with detectors given by Eq. (2). The non-CSS case is treated in Definition 4/S-A with detectors of the form (S.1). The authors claim applicability to stabilizer, subsystem, and dynamical codes, and give three examples: the qudit toric code, the [[5,1,3]] perfect qudit code, and a qudit generalization of the CSS honeycomb code. Under a simplified noise model—perfect preparation of the graph state followed by independent Pauli-Z errors—Proposition 3 maps the protocol to a circuit-based phenomenological noise model. Numerical HDRG-decoder thresholds for the foliated toric code are reported in Table I and claimed to increase with qudit dimension.","tokens_in":18379,"tokens_out":4214,"duration_ms":49091,"significance":"If the central claims are fully established, the paper would be a useful and nontrivial generalization of foliated MBQC to qudits, covering non-CSS and dynamical codes in a unified graph-state picture. The detector equations are derived from graph-state stabilizers rather than fitted to data, which is a strength, as is the concrete treatment of the toric-code unit cell and the non-CSS perfect-code example. The public simulation data and use of the sdim stabilizer simulator are also positive features. However, the advertised 'fault-tolerant' MBQC result is only demonstrated under a very restricted noise model, and the generality for arbitrary non-CSS/dynamical codes is asserted rather than proved. These issues are load-bearing for the paper's main claims, though they appear fixable by adding proofs and by qualifying the fault-tolerance statement.","major_comments":[{"comment":"The claim that the construction gives 'fault-tolerant measurement-based quantum computing' is not supported as stated. Proposition 3 and the simulations assume the graph state Lambda is prepared perfectly and only Pauli-Z errors occur afterwards. This excludes photon loss, failed fusion, and errors during the CZ entangling operations that are the dominant noise mechanisms for the photonic platform invoked throughout. The conclusion itself acknowledges that a photonic implementation would require joining smaller pieces via destructive fusion and defers that to future work. Please either prove fault tolerance under a preparation/loss-inclusive noise model or explicitly restrict the claim to phenomenological Z-noise after perfect preparation.","section":"Abstract; Conclusion; Appendix A"},{"comment":"The central claim that the framework applies to 'any Pauli-based code' is not fully proved for non-CSS and dynamical codes. Proposition 3 is stated and proved only for the CSS case of Definition 1. For non-CSS codes, Definition 4 gives a construction and Eq. (S.1) states a detector form, but no general theorem demonstrates that these are stabilizers of the graph state commuting with all the prescribed measurements for arbitrary check matrices r(k), nor how the dynamical-case detectors are constructed in general. The examples are not a substitute for a general proof. Add a rigorous argument, or state precisely which classes are actually proven.","section":"Definition 4; S-A; Eq. (S.1)"},{"comment":"The qudit CSS honeycomb code is presented as a new generalization of [26], but the paper offers only a one-sentence commutativity check ('This change ensures...'). Since this is the only dynamical-code example supporting the framework's claimed scope, the authors should verify that the period-6 schedule defines a valid Floquet code: the instantaneous stabilizer group, the deterministic measurement of the face stabilizers on the claimed rounds, and the logical operator structure. Without this, the dynamical-code example is not fully grounded.","section":"Appendix C; S-D E"},{"comment":"The threshold comparison is internally inconsistent. The abstract and conclusion say the foliated toric-code thresholds are 'comparable' to the non-foliated version, while Appendix B says they are 'significantly higher' and attributes the difference to a modified decoder. The comparison is also not controlled: it compares a rotated toric code to the unrotated surface-code results of [18] with a different HDRG implementation. Please either provide a matched comparison under identical decoder and code geometry, or temper the claim to avoid overstating the numerical validation.","section":"Appendix B; Table I"}],"minor_comments":[{"comment":"In Definition 4, 'r(t)' appears where the measurement round index k is meant; also the data-qudit index runs 0 to N-1 in Eq. (2)/(S.1) but 1 to N in Definition 1. Please reconcile the notation.","section":"Definition 4; S-A"},{"comment":"Typos: 'readers unfamilar' in Appendix B; 'necesssary' in S-D E. These should be corrected.","section":"Appendix B; S-D E"},{"comment":"Time-boundary detectors are left as 'a straightforward exercise for the reader.' Since the memory-experiment simulations and the claimed equivalence in Proposition 3 depend on boundary initialization/terminal measurement conditions, these should be specified explicitly for a self-contained treatment.","section":"Definition 2; main text"},{"comment":"The figures omit t=2 nodes. The captions should state clearly whether those nodes are absent from the graph or merely not shown, to avoid ambiguity about the graph-state structure used for the detector.","section":"Fig. 3(c) and Fig. S.10"}],"recommendation":"major_revision","confidential_remarks":"I see no signs of unethical practice; the concerns are about overclaiming the fault-tolerance result and missing general proofs for the non-CSS/dynamical cases. The mathematical core is plausible and likely publishable after the claims are either proved or appropriately restricted. I would encourage the editor to send it back with the request to add the missing proofs and to make the fault-tolerance statement match the actual noise model."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The foliation construction in [14,15] was qubit-only; this paper does the prime-dimensional qudit case properly. The main technical trick—alternating edge weights between layers to cancel F^2 != I—is exactly the right fix, and it is worked out in enough detail to be checked. The qudit CSS honeycomb code is a real new example, and the detector derivations in the main text and supplement are explicit, not hand-waved. The toric-code thresholds, while under a simple noise model, at least show the construction has a working decoding graph.\n\nSoft spots, in rough order of size. First, the fault-tolerance claim in the abstract is wider than what is proven. Appendix A establishes equivalence to circuit-based phenomenological noise only for perfect graph-state preparation followed by Z errors. Photon loss, fusion failures, and errors during entangling operations are exactly the errors a photonic qudit MBQC proposal needs to address, and they are outside the model. The conclusion is honest about this (FBQC for qudits is left to future work), so the right fix is to soften the abstract and make the noise-model boundary explicit early.\n\nSecond, the universal claim \"any Pauli-based code\" is plausible but not proven as a single theorem. Definition 4 and the detector formula S.1 are given for the non-CSS case, and the dynamical case is described verbally. For a paper with this scope, a unified statement with a proof that all stabilizer products yield detectors would remove the gap. As is, the worked examples carry the argument, and they are convincing, but a skeptic can reasonably ask for the general proof.\n\nThird, the numerics. The modified HDRG decoder (minimum-spanning-tree choice in step 3) is acknowledged by the authors to possibly improve on [18], so the \"comparable to non-foliated\" wording is misleading—the comparison is not like-for-like, and the higher thresholds may be a decoder artifact. No error bars are given. This is a minor issue for the central construction, but it should be flagged in any review.\n\nLast minor point: boundary detectors are left as an exercise, which is fine in a first pass but should eventually be spelled out.\n\nOverall: this is a solid theoretical construction that deserves a serious referee. The central graph-state construction and detector algebra look right; the main work for the authors is to align the claims with the noise model and to tighten the general proof.\n\nRecommendation: send to peer review. I would cite it.","headline":"A genuine, non-trivial qudit extension of foliation with an honest but restrictive noise model; the abstract overclaims fault tolerance for photonic platforms.","tokens_in":18811,"tokens_out":1894,"would_cite":true,"duration_ms":19981,"reading_group":"yes","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 claims that every Pauli-based quantum error-correcting code over prime-dimensional qudits can be foliated into a graph state whose X measurements give fault-tolerant measurement-based quantum computing.","keywords":["qudit","foliation","measurement-based quantum computation","graph states","Pauli codes","quantum error correction","qudit toric code","dynamical codes"],"falsifier":"Test the foliated qudit toric code with a noise model that adds preparation errors (e.g., depolarizing noise or photon loss during graph-state growth) while using the identical decoding graph; if the logical error rate does not drop below the physical error rate as code distance increases, the practical fault-tolerance claim fails.","tokens_in":18000,"feed_emoji":"⚛️","tokens_out":6030,"duration_ms":58569,"temperature":0.7,"pith_summary":"The paper establishes a general recipe: starting from any quantum error-correcting code whose checks are Pauli operators over a prime-dimensional qudit, build a layered graph state whose single-qudit measurements reproduce the code's syndrome extraction round by round. The construction covers CSS stabilizer codes, non-CSS stabilizer codes such as the [[5,1,3]] perfect code, and dynamical codes such as a new qudit generalization of the CSS honeycomb code. The point is that one can perform fault-tolerant measurement-based quantum computing in higher-dimensional systems—potentially useful in photonics, where qudit states are easy to prepare and losses dominate—without designing a new code for each hardware platform. If the construction is correct, error detection is built into the graph state: certain products of measurement outcomes, called detectors, are guaranteed to be trivial unless errors occurred. Simulations of the foliated qudit toric code show a threshold that increases with qudit dimension and is comparable to or better than circuit-based implementations.","feed_headline":"Foliation maps every qudit Pauli code to a fault-tolerant graph state","feed_subtitle":"Two definitions plus a detector formula extend the surface-code trick to CSS, non-CSS, and dynamical codes in any prime dimension.","key_machinery":"The central object is the foliated qudit graph state Λ (Definition 1 for CSS codes, Definition 4 for general Pauli codes): one linear chain state per data qudit, with CZ edges of alternating sign (−1)^t between consecutive time layers to cancel the unwanted Fourier factors of qudit teleportation, plus ancilla nodes connected to data qudits with weights given by the code's check matrices. The load-bearing identity is the detector D(c,t) = X_(1,c,t) X^{-1}_{(1,c,t+2)} ∏_q (X^{(H(t))_{cq}}_{(0,q,t+1)})^{(-1)^t}, which is a stabilizer of the graph state that commutes with every measurement; multiplying elementary detectors yields detectors for subsystem and dynamical codes. For non-CSS codes, an","core_discovery":"The paper's central claim is that any Pauli-based code over prime-dimensional qudits admits a foliation: a graph state Λ(d,N,T,H) whose nodes are data qudits arranged in chains (one per code qudit, with alternating CZ±1 edges) and ancilla qudits that entangle with the supports of the code's checks at each time step. Measuring X on every data qudit and the appropriate Pauli on each ancilla teleports the logical state forward in time while extracting syndrome information. The crucial identity is the detector D(c,t): a product of the same ancilla's measurements two time steps apart, together with powers of X on the intermediate data qudits, chosen so that all Z factors cancel via the alternatin","pith_inferences":["The authors note the equivalence only under a noise model with perfect graph-state preparation and post-preparation Pauli-Z errors. A natural extension is to analyze preparation errors, loss, and fusion failures in qudit photonic settings; until then, 'fault-tolerant' should be read as conditional.","The detector construction is algebraic and likely carries over to non-prime dimensions or finite fields with adjusted Weyl operators, though the paper does not claim this.","The reported toric-code threshold improvement may partly reflect the decoder heuristic's minimum-weight spanning-tree tie-breaking rather than the foliation itself; a head-to-head decoder comparison would isolate the source.","The layer-by-layer resource view suggests a testable extension: generate graph states sequentially with deterministic emitters and verify that detector failure rates scale with layer size and code distance as the threshold curves predict."],"forward_implications":["Every qudit stabilizer, subsystem, and dynamical Pauli code in prime dimension has a measurement-based implementation with the same syndrome structure as its circuit version, so error-correction results transfer directly.","Qudit MBQC can exploit higher-dimensional codes: the foliated toric-code threshold rises with d, from about 0.023 at d=3 to about 0.088 at d=7919 in the simulated noise model.","The construction covers non-CSS codes by adding ancilla–ancilla edges and adjusting ancilla measurement bases, so it is not limited to CSS codes.","The qudit CSS honeycomb code generalization gives a Floquet example whose detectors span five time steps, showing that dynamical codes can be foliated.","Because only 2–3 layers of the graph state need be present at once, the protocol is adaptable to photonic generation and fusion, and the paper frames qudit fusion-based quantum computation as the next step."],"fun_headline_variants":["Foliate any qudit Pauli code to a graph state","Every qudit Pauli code has a foliation","Qudit codes foliate to measurement-based QC","Foliation enables fault-tolerant qudit MBQC","Prime-dim qudit codes become graph states"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The fault-tolerance guarantee holds only when the graph state is prepared perfectly and all later noise is Pauli-Z; errors during graph-state preparation, such as photon loss or fusion failure, are not covered.","fun_headline_variants_meta":{"raw":{"variants":["Foliate any qudit Pauli code to a graph state","Every qudit Pauli code has a foliation","Qudit codes foliate to measurement-based QC","Foliation enables fault-tolerant qudit MBQC","Prime-dim qudit codes become graph states"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00022,"raw_usage":{"total_tokens":1261,"prompt_tokens":698,"completion_tokens":563,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":442,"completion_tokens_details":{"reasoning_tokens":488}},"tokens_in":442,"tokens_out":563,"duration_ms":6228,"temperature":1.0,"reasoning_tokens":488,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T03:42:55.988443+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the foliated qudit toric code with a noise model that adds preparation errors (e.g., depolarizing noise or photon loss during graph-state growth) while using the identical decoding graph; if the logical error rate does not drop below the physical error rate as code distance increases, the practical fault-tolerance claim fails.","supporting_citations":[],"review_version":1}