{"id":"f78bfca3-d645-4200-9cd9-9b4dde5b1069","arxiv_id":"2412.14288","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Topological and fracton ordered states can serve as perfect resources for nonlocal quantum games by encoding GHZ-like measurement statistics in braiding operators.","lead":"This paper shows that states from several exotic phases of quantum matter, such as the three-dimensional toric code, the X-cube fracton model, and the double-semion model, can be used as shared resources to win certain coordination games with certainty. It ties the winning strategies to the braiding behavior of the phases' quasiparticles and generalizes the games to quantum error-correcting codes.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Robustness for the new X-cube and double-semion games is asserted rather than shown: only fixed-point strategies are checked, and the bridge to deformed states is an unproven higher-form/subsystem symmetry survival expectation that Sec. VII itself admits is not proven.","rationale":"The reader's weakest-assumption identification is also the most load-bearing point in my reading: the paper's headline is about robust, generic advantage, but the actual proofs are at fixed points. I checked the fixed-point constructions and found no internal inconsistency in the stabilizer relations or in the cellulation-game derivation; the deficiency is the unproven step from ideal codewords to perturbed resource states. Section III C contains the only robustness argument, and it is explicitly an expectation about higher-form symmetry survival. Section VII contains a direct concession that the general robustness claim is not proven. This is not a stylistic objection: without that bridge, the abstract's claim that 'robust advantage is a generic property' of topological and fracton phases is unsupported for every example except the previously studied 2D toric code. The proposed check — a perturbed-state calculation of the X-cube game — directly targets that bridge. If it shows O(1) advantage at fixed nonzero λ, the concern is substantially mitigated for the new fracton example; if it fails, the paper's central generalization collapses to fixed-point constructions plus a conjecture. I do not see grounds for rejection, because the fixed-point results and the cellulation-game family are valuable and appear correct; a conditional verdict asking either for the robustness calculation or for softened claims is appropriate. I agree with the reader rather than raising a separate internal-correctness objection, because no concrete error in the stabilizer or game algebra surfaced in my review.","tokens_in":21547,"tokens_out":11178,"duration_ms":114195,"concrete_test":"Compute the average winning probability p_q(λ) for the three-player X-cube game using the operators in Eq. (22), with the resource state taken as the ground state of H_XC + λ Σ_e (h^X_e X_e + h^Z_e Z_e) on small periodic lattices (e.g., L = 3, 4, 5, with degeneracy resolved consistently), using exact diagonalization or a cluster-expansion/tensor-network method. Evaluate p_q via Eq. (13) with M_3 constructed from (22) and compare to the classical bound 3/4. The robustness bridge is supported if there is a fixed λ > 0, independent of L, for which p_q(λ,L) ≥ 3/4 + c with c > 0; it is refuted if p_q approaches 3/4 as L grows at any fixed nonzero λ. The same perturbed-state check on the double-semion generalized magic-square game with a local perturbation of H_DS would settle whether the second unverified example is robust.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The fixed-point constructions appear internally consistent: the 3D toric code operators in Sec. IV A, the X-cube operators in Sec. IV B, the double-semion operators in Sec. VI, and the cellulation-game argument in Sec. V B all give perfect strategies for the relevant ideal resource states, and Eq. (28) correctly evaluates to p_q = 1 for codewords. The paper's central claim, however, is not that ideal fixed points win perfectly but that topological and fracton phases generically provide a robust and scalable resource. That claim rests on the statement in Sec. III C that p-form symmetries with p > 0 'are expected to survive in the sense that the symmetry operators may change but the group structure is approximately preserved in the low-energy subspace' — an expectation, not a theorem. No perturbed winning probability is computed for any of the new examples. For the X-cube game the relevant symmetry is a subsystem/planar symmetry rather than a p-form symmetry, and for the double-semion game the magic-square strategy uses qudit Bell-like operators whose validation uses only the fixed-point constraints U_i \\tilde U_i^\\dagger = 1. The paper's own Sec. VII concedes that 'we have not proven' that every topological or fracton phase has a corresponding game, and phrases the deformed-state question as future work. Thus the generic robustness headline extrapolates from the 2D toric-code case of Ref. [16] to phases whose games are only verified at fixed points.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs explicit operator strategies for several nonlocal games using fixed-point ground states of the 2D and 3D toric code, the X-cube model, and the double-semion model, and it introduces a general family of 'cellulation games' for which codewords of d-dimensional homological codes give perfect strategies. It presents these constructions as evidence that topological and fracton ordered phases generically provide a robust and scalable resource for nonlocal quantum games, with braiding statistics as the source of contextuality. The fixed-point strategies are checked through stabilizer and braiding arguments, and the order/disorder parameter picture is used to unify them. The robustness of the new games beyond the fixed point is asserted rather than proven.","tokens_in":21775,"tokens_out":6525,"duration_ms":62761,"significance":"If the fixed-point constructions are correct, the paper significantly broadens the known family of quantum pseudo-telepathy resources beyond GHZ states and the 2D toric code, and the cellulation-game framework gives a clean homological unification. The explicit operator arrangements are parameter-free and checkable, and the homological counting in Eqs. (23) and (28) is a genuine contribution. The main advertised advance, however, is the claim of generic robust quantum advantage; for all new examples this robustness is not demonstrated, and the paper itself concedes that the general statement is unproven.","major_comments":[{"comment":"The abstract and Sec. I claim that topological and fracton ordered states 'in general provide a robust and scalable resource,' but for all new examples robustness is asserted rather than demonstrated. For the 2D toric code, robustness is inherited from Ref. [16]; for the 3D toric code, X-cube, and double-semion games, the paper verifies only fixed-point constraints [e.g., Eqs. (16), (18), (22), and the constraints X1 Xtilde1 = 1, Z1 Ztilde1^dagger = 1 in Sec. VI] and never evaluates the winning probability p_q for a deformed state. The only bridge offered is the statement in Sec. III C that p-form symmetries with p>0 'are expected to survive,' which is an expectation, not a theorem, and Sec. VII explicitly concedes that no general proof is provided. This point is load-bearing because the advertised generic robust advantage is precisely what distinguishes the paper from a collection of fixed-point perfect strategies. I request either a proof for the new examples (e.g., a lower bound on the relevant Mermin expectation under local perturbations) or a substantial revision that presents the constructions as fixed-point perfect strategies and leaves robustness as a conjecture.","section":"Sec. III C and VII"},{"comment":"The X-cube game is built from prism and cage operators that implement lineon-fracton braiding through subsystem (planar) symmetries, not p-form symmetries. Consequently, even if the Sec. III C expectation about higher-form symmetries were proven, it would not cover this example. The robustness of the X-cube strategy requires a separate argument about subsystem symmetries and their behavior under local perturbations; no such argument appears in Sec. IV B.","section":"Sec. IV B"},{"comment":"The double-semion game is verified only at the fixed point, using the exact stabilizer constraints X1 Xtilde1 = 1 and Z1 Ztilde1^dagger = 1 together with the generalized commutation relations Z_i X_j = i^{delta_ij} X_j Z_i. The paper does not discuss how these constraints behave away from the fixed point, and no perturbed winning probability is computed. Because the resource here involves qudit Bell-like operators and self-statistics, the p-form symmetry robustness narrative of Sec. III C does not directly apply. This leaves the robustness claim for the double-semion game unsupported.","section":"Sec. VI"}],"minor_comments":[{"comment":"In the paragraph following Eq. (28), 'handed handed to each player' contains a duplicated word.","section":"Sec. V B"},{"comment":"The statement that player A receives an integer x_A in {01,10,11} is confusing: 01, 10, and 11 are binary strings, not integers. Please write them as elements of {0,1}^2 or define the intended integer values explicitly.","section":"Sec. VI"},{"comment":"The footnote says phase estimation is exact in the present setting and cites Ref. [59], but Ref. [59] is a paper on topological proofs of contextuality, not a phase-estimation reference. Please correct the citation or add the intended reference.","section":"Footnote [55]"},{"comment":"The sentence 'Open strings of Z_f operators on the dual lattice end on cube centers' is hard to follow because the Z operators in Eq. (14) live on faces. Please clarify the duality convention used for this statement so the reader can track which lattice carries the string operators.","section":"Sec. IV A"}],"recommendation":"major_revision","confidential_remarks":"The fixed-point constructions appear sound and the cellulation-game framework is a nice contribution, but the title and abstract promise robust generic advantage that the new examples do not yet establish. The only rigorous robustness result imported from Ref. [16] concerns the 2D toric code. I would recommend major revision: either supply perturbed-state robustness calculations for the 3D toric code, X-cube, and double-semion games, or reframe the claims as fixed-point perfect strategies with robustness as an explicit conjecture."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The real content is the explicit constructions: perfect strategies for the 3D toric code (both 1-form and 2-form symmetries), X-cube, and double-semion, plus the cellulation-game family for homological codes. Those are new relative to Refs. [12,13,16], and they check out as far as I can see—the stabilizer constraints and braiding arguments are explicit, and Eq. (28) is a clean homological evaluation. The order/disorder unification is a genuinely useful perspective: it explains why the games are robust when the relevant symmetry is higher-form and almost-local, and why GHZ-like resources are fragile. The double-semion game using self-statistics is a nice extension beyond mutual statistics.\n\nThe soft spot is exactly what the stress test says. The abstract and intro claim robust advantage is generic for topological and fracton phases, but for the new games robustness is not demonstrated. No perturbed winning probability is computed for X-cube or double-semion; Sec. III C's expectation that p-form symmetries survive is an expectation, and Sec. VII explicitly says 'we have not proven' the general statement. The 2D toric code case from Ref. [16] carries the burden, and the new games are only verified at fixed points. That doesn't invalidate the constructions, but it means the headline should be read as a conjecture with strong support, not a theorem. The paper is honest about this in the discussion, so it's not a hidden flaw.\n\nOne more minor point: the X-cube game uses a subsystem/planar symmetry rather than a p-form symmetry, so the robustness rationale doesn't transfer automatically. The authors don't pretend otherwise, but it's worth a referee asking them to be precise about that.\n\nNet: this deserves a serious referee. The constructions are formal, reproducible, and likely useful for benchmarking topological order on hardware. The robustness claim should be softened or explicitly flagged as open, which the paper already half does. Recommend accept-with-revisions or, at minimum, careful review. I'd cite the cellulation games if I were working on homological codes.","headline":"A solid set of new game constructions for topologically ordered resource states, with a robustness headline that outruns the proof.","tokens_in":22391,"tokens_out":1767,"would_cite":true,"duration_ms":14983,"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":"Topological and fracton ordered phases are generic sources of robust quantum advantage, with braiding statistics supplying the contextuality.","keywords":["nonlocal quantum games","topological order","fracton order","braiding statistics","contextuality","quantum pseudo-telepathy","homological quantum codes","higher-form symmetries"],"falsifier":"Take a 3D toric code or X-cube Hamiltonian, add a small uniform perturbation such as $-h\\sum_e X_e$, prepare the ground state, and compute the players' average victory probability for the parity game; the robustness claim requires an $O(1)$ gap above the classical bound for all sufficiently small $h$, so observing that gap decay exponentially with system size for any fixed nonzero $h$ would falsify the central claim.","tokens_in":21293,"feed_emoji":"🎲","tokens_out":9187,"duration_ms":74785,"temperature":0.7,"pith_summary":"This paper argues that the robust quantum advantage at nonlocal games previously demonstrated for the 2D toric code is not an accident of that model: any state drawn from a topological or fracton ordered phase can be turned into a resource for a suitably designed game, with the braiding statistics of its excitations supplying the needed contextuality. The authors construct explicit perfect strategies using 3D toric code ground states and X-cube ground states for the multiparty parity game, and double-semion ground states for a generalized magic-square game, showing that both mutual and self-statistics can be harnessed. They unify these constructions by viewing the measured operators as order and disorder parameters of an underlying generalized symmetry-breaking transition, and they introduce a broad family of 'cellulation games' for which codewords of homological quantum codes in $d$ dimensions are perfect resources. The result matters because it turns phases of quantum matter into proof-of-principle computational assets that are robust to small deformations of the resource state, and suggests quantum games could be used to diagnose topological or fracton order on hardware.","feed_headline":"Braiding turns topological phases into perfect game resources","feed_subtitle":"3D toric code, X-cube, and double-semion states give quantum strategies that beat all classical ones and survive deformations.","key_machinery":"The carrying object is the twist product of string and membrane operators, together with an order--disorder parameter pair attached to a higher-form symmetry. For each player the strategy designates an $X$-operator, a truncated symmetry operator (a segment of a loop, membrane, or prism), and a $Z$-operator charged under the full symmetry, arranged so that $X_i$ and $Z_i$ intersect once, $X_1 X_2 \\cdots X_P = 1$, and $Z_i Z_j = 1$ on the ground state. The minus signs that make the collective measurement win are computed by the twist product, which interleaves the two operators so their noncommutation at the intersection is exposed; in the toric-code examples this is exactly the braiding phase of $e$ and $m$ anyons, in the X-cube model the phase from braiding lineons around fractons, and in the double-semion model the phase from exchanging $s$ anyons. Because the relevant symmetries are higher-form, the full symmetry operators can be chosen almost local, as small loops, membranes, or cages, so the measured quantities avoid the global orthogonality catastrophe that makes GHZ-based strategies fragile.","core_discovery":"On the paper's own terms, the central claim is that topological and fracton ordered states in general provide a robust and scalable resource for nonlocal quantum games, with braiding statistics as the source of contextuality. Concretely: ground states of the 3D toric code and the X-cube fracton model admit perfect strategies for the $P$-player parity game, based respectively on 1-form/2-form symmetries and on lineon--fracton braiding; ground states of the double-semion model admit a perfect strategy for a generalized magic-square game, based on anyon self-statistics; and for any cellulation of $d$-dimensional space, codewords of a homological CSS code give a perfect strategy for the associated cellulation game using only single-site Pauli measurements. The previously studied 2D toric-code game is recovered as the special case where the embedded state is a three-qubit GHZ state. These are constructive existence results: the measured operators are chosen so that their collective product is a stabilizer of the resource state up to the phase required by the game's victory condition.","pith_inferences":["Testable extension: applying the same order--disorder construction to twisted quantum doubles or to type-II fracton models obtained by fractalization should produce additional perfect strategies; the paper mentions fractalization as a route but does not work out the games.","If the robustness expectation is borne out, the games function as an operational order parameter for topological order, detecting the phase without needing to measure topological entanglement entropy or anyonic data directly; the paper only frames games as a diagnostic, not as an equivalent order parameter.","The cellulation games suggest that two phases with different braiding data will generally be distinguishable by the set of games they can win with certainty, raising the possibility of a game-theoretic classification of topological order; the paper poses this as an open question rather than a claim.","The double-semion example indicates that chiral or non-Abelian phases, whose statistics include nontrivial self-exchanges, should also admit nonlocal games, but the paper does not construct them."],"forward_implications":["For every $P \\geq 3$, the 3D toric code and X-cube strategies win the parity game with certainty, while any classical strategy wins on at most $\\frac12 + \\frac{1}{2\\lceil P/2\\rceil}$ of inputs.","The double-semion game shows that self-statistics alone can power a perfect strategy, so the construction is not limited to mutual braiding between distinct particle types.","If the higher-form symmetry survival expectation holds, the $O(1)$ advantage persists across the 3D toric code, X-cube, and double-semion phases, meaning deformed, experimentally prepared states rather than idealized fixed points remain useful resources.","The cellulation-game construction yields a perfect strategy for any homological CSS code in $d$ dimensions, making the resource property a generic feature of such codes rather than a special property of the toric code.","These games provide a Hamiltonian-independent route to diagnosing topological or fracton order: a state that wins the relevant game with $O(1)$ advantage is in the ordered phase, and one that does not is not."],"supporting_citations":[{"why":"Supplies the 2D toric code parity-game strategy and its robustness to deformation, which this paper generalises to other phases.","marker":"[16]"},{"why":"Provides an alternative multiplayer game for the toric code that the paper compares to and extends with local strategies.","marker":"[12]"},{"why":"Defines the toric code Hamiltonian and anyon braiding that underpin the resource states.","marker":"[27]"},{"why":"Defines the X-cube model and its fracton/lineon excitations used for the fracton-game strategy.","marker":"[39]"},{"why":"Introduces the double-semion string-net model whose self-statistics are harnessed in the magic-square game.","marker":"[48]"},{"why":"Supplies the twist product that produces the minus signs encoding braiding statistics.","marker":"[28]"},{"why":"Provides the order/disorder parameter language used to unify GHZ, toric code, 3D toric code, and X-cube strategies.","marker":"[29]"},{"why":"Gives the optimal classical bound for the multi-player parity game that the quantum strategies must beat.","marker":"[24]"},{"why":"Reviews higher-form symmetries used to explain robustness and to construct the 3D toric code and X-cube operators.","marker":"[33]"}],"fun_headline_variants":["Braiding statistics make topological states perfect at quantum games","Topological and fracton phases win quantum games via braiding","Perfect quantum strategies emerge from braiding in matter phases","Braiding gives topological states unbeatable quantum game advantage","How braiding turns phases of matter into winning quantum resources"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The bridge from fixed-point perfect strategies to the advertised robust advantage is the expectation, stated as unproven in Sec. III C, that $p$-form symmetries with $p>0$ survive small perturbations with their group structure approximately preserved in the low-energy subspace; if that expectation fails, the games remain perfect at exact fixed points but the generic robust-and-scalable-resource claim is unsupported.","fun_headline_variants_meta":{"raw":{"variants":["Braiding statistics make topological states perfect at quantum games","Topological and fracton phases win quantum games via braiding","Perfect quantum strategies emerge from braiding in matter phases","Braiding gives topological states unbeatable quantum game advantage","How braiding turns phases of matter into winning quantum resources"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000256,"raw_usage":{"total_tokens":1606,"prompt_tokens":1010,"completion_tokens":596,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":626,"completion_tokens_details":{"reasoning_tokens":517}},"tokens_in":626,"tokens_out":596,"duration_ms":5828,"temperature":1.0,"reasoning_tokens":517,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:22:31.528413+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a 3D toric code or X-cube Hamiltonian, add a small uniform perturbation such as $-h\\sum_e X_e$, prepare the ground state, and compute the players' average victory probability for the parity game; the robustness claim requires an $O(1)$ gap above the classical bound for all sufficiently small $h$, so observing that gap decay exponentially with system size for any fixed nonzero $h$ would falsify the central claim.","supporting_citations":[{"cited_title":"Castelnovo, C","cited_arxiv_id":null,"evidence_quote":"Introduces the double-semion string-net model whose self-statistics are harnessed in the magic-square game."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the 2D toric code parity-game strategy and its robustness to deformation, which this paper generalises to other phases."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the order/disorder parameter language used to unify GHZ, toric code, 3D toric code, and X-cube strategies."},{"cited_title":"Quantum tasks assisted by quantum noise","cited_arxiv_id":"2308.10969","evidence_quote":"Gives the optimal classical bound for the multi-player parity game that the quantum strategies must beat."},{"cited_title":"Brassard, A","cited_arxiv_id":null,"evidence_quote":"Reviews higher-form symmetries used to explain robustness and to construct the 3D toric code and X-cube operators."}],"review_version":1}