{"id":"b81636e3-9ec3-414f-b938-ff2e4e13d90a","arxiv_id":"2505.03889","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A noisy stabilizer formalism is generalized from qubits to prime-power qudits, with update rules for Pauli-diagonal noise under Clifford operations and Weyl measurements, demonstrated on Bell-pair generation from a noisy linear cluster state.","lead":"This paper introduces a formalism for tracking how Pauli-diagonal noise spreads when high-dimensional quantum systems called qudits are manipulated with Clifford operations and measurements. It lets researchers analyze noisy qudit protocols such as Bell-pair generation without storing exponentially large density matrices, as long as the final state is small.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Applicability to all stabilizer states in even prime-power dimensions rests on a cited but unproven local-Clifford equivalence; the paper should state and verify this theorem explicitly.","rationale":"The reader's weakest_assumption identifies the same load-bearing concern: the extension from graph states to all stabilizer states relies on a local-Clifford-equivalence theorem that is cited but not proven, with possible phase subtleties in even prime-power dimensions. The paper is otherwise honest about limitations (prime-power restriction, Pauli-diagonal noise, final-state size), and the update rules and application appear internally consistent. Because the theorem is standard in the qudit stabilizer literature and is likely valid for all prime powers, this concern does not warrant rejection; however, it justifies the CONDITIONAL verdict until the equivalence is stated explicitly and verified for even prime-power dimensions, or until the cited references are confirmed to cover precisely the construction used. No deeper mathematical flaw was found in the update rules or the fidelity derivation, so the reader's conditional verdict remains appropriate.","tokens_in":31464,"tokens_out":34820,"duration_ms":315553,"concrete_test":"Enumerate all stabilizer states for n=2 and n=3 qudits of dimension d=4 (and d=8 if feasible) under the even-prime-power construction of Appendix A, and check whether each is local-Clifford equivalent to a weighted graph state using the Clifford group generated by Table I with the Appendix A S gate. Alternatively, verify that Refs. [27,28,45,46] explicitly prove the equivalence for all even prime powers and that their phase conventions match those used in the paper; if the references cover this and the conventions align, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim extends the graph-state NSF_d to all stabilizer states in prime-power dimensions via the assertion in Sec. VI.A that every such stabilizer state is local Clifford equivalent to a graph state, citing Refs. [27,28,45,46]. The paper does not prove this equivalence, and for even prime-power dimensions the local Clifford group and graph-state definitions depend on the Galois-ring construction of Appendix A, where phase conventions (S gate, trace, lift) are non-trivial. If the theorem fails for some stabilizer states in, e.g., dimension 4, then the claimed applicability to 'stabilizer states in prime-power dimensions' is incomplete: those states could not be handled by the graph-state update rules, and the alternative direct-stabilizer method sketched in Sec. VI.A is itself only outlined and relies on the same restriction property. The graph-state formalism would survive, but the abstract's full generality would not. This is a citation/presentation gap rather than a demonstrated error, but it is load-bearing for the extension to general stabilizer states.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces a qudit generalization of the noisy stabilizer formalism (NSFd), extending the authors' earlier qubit formalism to prime-power local dimension d = p^m. The central idea is to treat Pauli-diagonal noise channels and stabilizer/graph states independently, updating the noise channels under Clifford operations and generalized Pauli (Weyl) measurements via a set of graphical update rules. The paper derives explicit graphical rules for Weyl measurements on graph states, noise update rules for the building-block Clifford operations, and an extension to general stabilizer states. It also discusses efficiency, giving a linear-in-initial-qudits and exponential-in-final-qudits scaling claim, and provides an extension to arbitrary finite dimensions via a linearized stabilizer formalism. As an application, the authors analyze the generation of a generalized Bell pair from a noisy linear cluster state under two depolarizing noise sources, deriving an analytic fidelity formula and a dimension-dependent noise parameter q_d fixed by matching Choi-Jamiołkowski fidelities.","tokens_in":31629,"tokens_out":12467,"duration_ms":129397,"significance":"If correct, the formalism is a useful analytical tool for noisy qudit protocols whose final state is small, extending the established qubit noisy stabilizer formalism to prime-power qudits. The paper is strong in its explicit derivations: Appendices B and C give detailed proofs of the measurement and noise update rules, and Appendix E provides a complete analytic fidelity formula. The dimension-dependent noise parameter q_d is fixed by matching Choi fidelities of qubit channels rather than fitted to the final fidelity, which is a methodological strength. The main risk concerns the breadth of the claimed applicability: the extension to all stabilizer states in prime-power dimensions rests on a cited but unproven local-Clifford equivalence, and the efficiency statement in Sec. V is not fully precise.","major_comments":[{"comment":"The extension of the formalism from graph states to all stabilizer states in prime-power dimensions rests on the statement that every such stabilizer state is local Clifford equivalent to a graph state. The manuscript cites Refs. [27,28,45,46] but neither states the theorem precisely nor proves it for even prime-power dimensions, where Appendix A introduces different phase conventions (e.g., the S gate and H_even) and a Galois-ring construction. Since this is the only argument supporting the abstract's claim of applicability to all stabilizer states in prime-power dimensions, the authors should state the exact theorem and either provide a proof or give a reference that explicitly covers all p^m, including p=2 and m>1. If no such reference covers the even case, the applicability claim must be restricted to graph states or to the class for which the equivalence is proven.","section":"§VI.A, first paragraph; Appendix A"},{"comment":"The direct stabilizer method assumes that for every measurement vertex v there exist stabilizer operators S_i whose local parts {S_i^v} generate the full local Weyl group, 'unless vertex v is disconnected in the local Clifford equivalent graph.' This assertion is load-bearing for the claimed treatment of Weyl measurements on general stabilizer states, but it is not proved and the condition is not made precise. The authors should provide a proof or a precise statement of this stabilizer-generation property, including the even-prime-power case, so that the direct method is more than a sketch.","section":"§VI.A, paragraph beginning 'Alternatively, one can directly work with stabilizer states'"},{"comment":"The complexity claim that the formalism 'scales linearly with the number of qudits in the initial state' is not stated with enough precision to be checked. The text says the number of different update rules is at most d×n, but for n single-qudit Pauli channels one has d^2×n noise terms, and each term is generally a product of elementary Z operators that must be combined. Please state the complexity of the full procedure with an explicit formula, including the d^2 factor and the cost of composing elementary updates, and specify whether d is treated as a constant or as an asymptotic parameter. The current wording conflates the number of elementary update rules with the number of noise terms and leaves the advertised exponential-in-final-size scaling incompletely specified.","section":"§V and abstract"},{"comment":"The claim that the formalism 'remains applicable' for arbitrary finite dimensions is qualified in Appendix D by the condition that the Clifford equivalences for projectors do not require multiplicative inverses of zero-divisors, but the manuscript does not say which dimensions or which Weyl measurements satisfy this condition. Please specify the exact class of operations for which the arbitrary-dimension update rules are proven, and give at least one explicit example of a composite dimension and measurement where the restriction bites. Without this, the arbitrary-dimension part of the abstract is stronger than what the appendix actually establishes.","section":"§VI.B and Appendix D"}],"minor_comments":[{"comment":"The block-vector notation 0_p and e_j is not defined, and e_j collides with the standard unit vector e_v introduced in Sec. II.D. Please define the index range of j and the length of each block explicitly.","section":"Eq. (29)"},{"comment":"The sentence 'A′ is the adjacency matrix of τ_{w0}(q)(G)' uses an undefined symbol q; it should presumably be r as in Eq. (17).","section":"§III.D, paragraph after Eq. (17)"},{"comment":"The sentence defining the weight-vector entries with the function H(x)=1 for x≤0 and H(x)=0 for x>0 is easy to misread. Rewrite the entries w^k_1 directly as m + [k≤s] using Iverson brackets, which is what Eq. (29) appears to encode.","section":"Appendix E3"},{"comment":"The definition of the set U_h via tuples (i^j_{k_1},...,i^j_{k_h}) is not clear. Please rewrite the index set using standard set notation, since the final fidelity formula depends on this function π_h.","section":"Appendix E2, step 4"},{"comment":"The use of the symbol '9' to denote subtraction makes several equations difficult to read; replacing it with the standard minus sign would improve clarity.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"This is an incremental but potentially sound extension of the authors' earlier qubit NSF [41]. The main technical risk is the unproven local-Clifford equivalence between stabilizer states and graph states in even prime-power dimensions; if the authors can supply a precise reference covering p=2, m>1, the paper may be acceptable after revision. The novelty is modest, but the explicit update rules and the worked Bell-pair application give it sufficient value as a methods paper for a quantum-information journal. I would recommend asking for the theorem statement and proof/reference for the even case, and a precise efficiency statement, rather than new physical results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the Aigner–Mor-Ruiz–Dür paper. The take-home: it is a real, workmanlike generalization of the qubit noisy stabilizer formalism to prime-power qudit graph states, and the core math is in good shape. The genuinely new parts are the graphical rules for local Weyl measurements on qudit graph states and the accompanying noise-operator update rules; both are derived in detail in the appendices, including the awkward even-prime-power phase conventions via Galois rings. The linear-cluster-to-Bell-pair application yields a fully analytical fidelity formula, and the complexity claim (linear in initial qudits, exponential in final qudits) is correct because the final density matrix sets the cost. I did not find a load-bearing error.\n\nThe biggest soft spot is one the authors mostly leave implicit. The expansion from graph states to all prime-power stabilizer states relies on the theorem that every stabilizer state is local Clifford equivalent to a graph state, cited to [27,28,45,46] but not proved or stated with the right hypotheses. For odd dimensions this is standard; for even prime powers, the Galois-ring phase conventions in Appendix A make it worth stating and checking explicitly. If the equivalence fails in, say, dimension 4, the graph-state formalism survives but the abstract's full generality would not. I regard this as a presentation/citation gap, not a demonstrated error, and it should be fixable in revision.\n\nMinor issues worth fixing: the basis vector e_j in Eq. (29) collides with the e_v notation used for vertex basis vectors; the complexity sentence in Sec. V ('d x n update rules') is loose; and Sec. VI.B attributes the linearized stabilizer formalism to [41], while Appendix D correctly cites [23]. These do not affect the mathematics.\n\nThe paper is honest about its own limitations: it stays within Pauli-diagonal noise, notes the exponential final-state cost, and is candid that arbitrary composite dimensions only get a restricted graph-state version. That matters and counts in its favor.\n\nWho should read it: anyone doing analytical noise analysis for qudit graph-state protocols — repeaters, purification, measurement-based qudit computation. It deserves a serious referee; I would send it out. With a small revision on the stabilizer-equivalence statement and the cleanups above, it should be acceptable.","headline":"A genuine generalization of the qubit NSF to prime-power qudit graph states with sound appendix derivations; the only real soft spot is the unproven (but standard) stabilizer-to-graph equivalence for even prime powers.","tokens_in":32174,"tokens_out":4407,"would_cite":true,"duration_ms":43053,"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":"Pauli-diagonal noise on qudit stabilizer states can be updated analytically under Clifford operations and generalized Pauli measurements, with cost linear in the initial state and exponential only in the final state.","keywords":["qudit","noisy stabilizer formalism","graph states","Pauli-diagonal noise","Weyl measurements","Clifford operations","depolarizing noise","generalized Bell pair"],"falsifier":"Run the update rules on a small system, for example three qudits of dimension $d=4$ in a linear cluster with depolarizing noise, and compare the final Bell-pair fidelity against full density-matrix evolution for the same measurement sequence; any mismatch at nonzero noise would falsify the update rules. Separately, exhibiting a $d=4$ stabilizer state that is not convertible to a graph state by single-qudit Clifford gates would falsify the claimed extension to all stabilizer states.","tokens_in":31246,"feed_emoji":"⚛️","tokens_out":7893,"duration_ms":74811,"temperature":0.7,"pith_summary":"The paper introduces the qudit noisy stabilizer formalism, a bookkeeping scheme for stabilizer states in prime-power dimensions that are acted on by generalized Pauli-diagonal noise and then processed by Clifford gates and generalized Pauli measurements. The central move is to track the noise channels separately from the pure stabilizer part, so the full density matrix never has to be written down. The cost grows linearly with the number of initial qudits and noise terms, and exponentially only with the number of qudits left at the end, which makes protocols that distill or teleport a few entangled qudits analytically tractable. The authors demonstrate this by computing the fidelity of a generalized Bell pair produced from a noisy linear cluster state under two noise sources.","feed_headline":"Noisy qudit states tracked without large density matrices","feed_subtitle":"Noise is updated separately from the state, so only the final few qudits set the cost.","key_machinery":"The central object is a set of update rules: for a manipulation operator $O$ and a $Z$-type noise term $N$, $O N |G\\rangle = \\tilde N O |G\\rangle$, with $\\tilde N$ determined graphically. The rules are assembled from three building blocks—local multiplication, local complementation, and the $Z$ measurement—and every Weyl measurement reduces to these because its projector is Clifford equivalent to the $Z$ projector. The graphical measurement rules for $Z$, $Y$-type, $X$, $X(m)$, and $W(n,m)$ provide the graph transformations that accompany the updates. Global phases are dropped because the channels are Pauli-diagonal.","core_discovery":"For a graph state in dimension $d = p^m$, any generalized Pauli noise term can be rewritten, using the graph stabilizers, as a product of $Z$-type operators. Each such operator commutes with a Clifford gate or a Weyl measurement up to a determined update: the noise operator is replaced by another $Z$-type operator whose support is read off from the current graph. Iterating these updates after every manipulation gives the exact final noisy state, and because prime-power stabilizer states are local Clifford equivalent to graph states, the same rules describe noisy stabilizer states generally. The only exponential step is applying the final updated noise maps to the small noiseless final state, so the formalism is efficient whenever the protocol ends in few qudits.","pith_inferences":["A natural experimental test is to realize the same linear-cluster Bell-pair protocol on qudits of dimension 2, 4, and 8 with controlled depolarizing noise and check whether the predicted intermediate-dimension optimum appears.","The noise–state separation suggests a continuous-variable counterpart: with the same commutation structure carried over to symplectic phase space, the formalism could describe Gaussian noise on continuous-variable cluster states.","For repeater protocols, the weight vector becomes a design parameter: one can search over measurement orders for the one that concentrates noise on terms that do not affect the target fidelity, without rerunning the full protocol each time.","Because the method tracks the exact noisy state rather than sampling from it, it can certify noise thresholds for small entangled states in regimes where Monte Carlo stabilizer simulation would need many samples."],"forward_implications":["Protocols that end in a small entangled target—Bell-pair generation, entanglement swapping, purification—can be analyzed exactly under Pauli noise without constructing the full density matrix.","The fidelity of the generated generalized Bell pair is obtained as a closed analytic function of the measurement-order weight vector, the local dimension $d$, and the depolarizing parameter $\\lambda$, so parameter scans are immediate.","The order of Weyl measurements on neighbouring qudits changes the final noise pattern even though the noiseless measurements commute, so protocols must specify and optimize an order.","The same bookkeeping applies to all stabilizer states in prime-power dimensions, and to graph states in arbitrary finite dimensions under operations that preserve the graph-state form.","Depending on how the depolarizing parameter scales with dimension, the adapted final fidelity can be highest for $d=2$, for large $d$, or for an intermediate dimension, so high-dimensional encodings are not universally better."],"supporting_citations":[{"why":"Supplies the qubit noisy stabilizer formalism and its independent treatment of noise channels and states, which this paper generalizes to qudits.","marker":"[41]"},{"why":"Establishes graph states and the local Clifford equivalence of stabilizer states, the bridge that extends the graph-state update rules to all stabilizer states.","marker":"[27, 28]"},{"why":"The Gottesman-Knill theorem, used to argue that Clifford operations and measurements on the noiseless stabilizer part are efficiently trackable.","marker":"[24-26]"},{"why":"Shows arbitrary Pauli channels can be replaced by Pauli-diagonal ones without changing diagonal elements or channel fidelity, justifying the restriction to Pauli-diagonal noise.","marker":"[48]"},{"why":"Provides the trapped-ion qudit platform whose dimension-dependent error rates motivate the two-noise-source application.","marker":"[13]"},{"why":"Works out the translation between noise-update order and simultaneous measurement patterns for qubits, which the qudit extension follows.","marker":"[43]"},{"why":"Shows composite-dimensional stabilizer states are generally not Clifford equivalent to graph states, explaining the prime-power restriction.","marker":"[51]"}],"fun_headline_variants":["Qudit noise tracked without density matrices","Noisy qudit states handled with linear cost in input","Qudit stabilizer formalism makes noise tracking efficient","Exact noise updates for qudit graph states","Qudit noise simulation scales with initial qudits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The full-generality version of the formalism rests on the premise that every stabilizer state in prime-power dimension can be converted to a graph state by single-qudit Clifford gates; if that fails for some even prime-power dimensions, only the graph-state version remains valid.","fun_headline_variants_meta":{"raw":{"variants":["Qudit noise tracked without density matrices","Noisy qudit states handled with linear cost in input","Qudit stabilizer formalism makes noise tracking efficient","Exact noise updates for qudit graph states","Qudit noise simulation scales with initial qudits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000485,"raw_usage":{"total_tokens":2333,"prompt_tokens":825,"completion_tokens":1508,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":441,"completion_tokens_details":{"reasoning_tokens":1433}},"tokens_in":441,"tokens_out":1508,"duration_ms":12849,"temperature":1.0,"reasoning_tokens":1433,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:45:36.474788+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the update rules on a small system, for example three qudits of dimension $d=4$ in a linear cluster with depolarizing noise, and compare the final Bell-pair fidelity against full density-matrix evolution for the same measurement sequence; any mismatch at nonzero noise would falsify the update rules. Separately, exhibiting a $d=4$ stabilizer state that is not convertible to a graph state by single-qudit Clifford gates would falsify the claimed extension to all stabilizer states.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the qubit noisy stabilizer formalism and its independent treatment of noise channels and states, which this paper generalizes to qudits."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the trapped-ion qudit platform whose dimension-dependent error rates motivate the two-noise-source application."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Works out the translation between noise-update order and simultaneous measurement patterns for qubits, which the qudit extension follows."}],"review_version":1}