Pith. sign in

REVIEW 3 major objections 4 minor 70 references

Braiding for the win: Harnessing braiding statistics in topological states to win quantum games

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Topological and fracton ordered phases are generic sources of robust quantum advantage, with braiding statistics supplying the contextuality.

desk verdict A solid set of new game constructions for topologically ordered resource states, with a robustness headline that outruns the proof. read the letter →

arxiv 2412.14288 v1 pith:UJRKT7P5 submitted 2024-12-18 quant-ph cond-mat.stat-mechcond-mat.str-el

Oliver Hart , David T. Stephen , Dominic J. Williamson , Rahul Nandkishore This is my paper · ORCID
classification quant-phcond-mat.stat-mechcond-mat.str-el
keywords nonlocalquantumgamestopologicalorderfractonbraidingstatisticscontextualitypseudo-telepathyhomologicalcodeshigher-formsymmetries
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Sec. III C and VII] 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.
  2. [Sec. IV B] 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.
  3. [Sec. VI] 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.
minor comments (4)
  1. [Sec. V B] In the paragraph following Eq. (28), 'handed handed to each player' contains a duplicated word.
  2. [Sec. VI] 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.
  3. [Footnote [55]] 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.
  4. [Sec. IV A] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: every fixed-point strategy is verified by direct stabilizer and braiding computation, and the robustness gap is an openly acknowledged assumption rather than a circular derivation.

full rationale

The derivation chain for the perfect strategies is self-contained. For the parity game, the quantum strategy is checked against standard GHZ stabilizer constraints in Sec. II. For the 2D toric code, the operators in (10) are chosen so that the constraints (11) hold by evaluation of twist products in fixed-point ground states, as shown in Sec. III B. The 3D toric code constructions (16) and (18) and the X-cube construction (22) are all verified by explicit stabilizer identities X1X2X3 = 1 and Zi Zj = 1; no fitted parameter or imported uniqueness result enters. The double-semion game in Sec. VI is validated against the fixed-point constraints U1 U1^dagger = 1, U2 U2^dagger = 1, and U3 U3^dagger = -1, which follow from A_v = 1 and standard string-net anyon data, and the operators (35) have the required qudit commutation relations by explicit check. The cellulation-game result is a direct computation: Eq. (27) evaluates the product of measured operators to a phase times stabilizers, so Eq. (28) gives p_q = 1 for codewords; this is a construction of a game for a resource, not a derivation that renames the resource property. The one self-citation, Ref. [16], is used as prior evidence for robustness in the 2D toric code case and is backed by trapped-ion experiments; it is not the argument for any of the new fixed-point constructions. The paper itself flags the limits of its robustness claim: Sec. VII states that 'while our results strongly suggest that any nontrivial topological or fracton ordered phase should have a corresponding nonlocal game at which it provides robust advantage, we have not proven this,' and Sec. III C describes higher-form symmetry survival as an expectation ('p-form symmetries with p > 0 are expected to survive') rather than a theorem. These are unsupported extrapolations that weaken the headline claim of generic robust advantage, but they are not circular reductions, because the fixed-point statements and the game constructions do not depend on them.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No numerical parameters are fitted; the constructions are parameter-free and use standard properties of stabilizer models, planar dual graphs, and dual cellulations. The main nonstandard input is the physics expectation that higher-form symmetries survive perturbations, which underlies the robustness claim.

assumptions (4)
  • domain assumption For p-form symmetries with p>0, explicit symmetry-breaking perturbations approximately preserve the group structure in the low-energy subspace, allowing order and disorder operators to survive.
    Invoked in Sec. III C to argue that robustness transfers from the 2D toric code to other topological and fracton phases. This is an expectation, and Sec. VII concedes the general robustness statement is not proven.
  • standard math A plane graph and its geometric dual provide dual loop operators with exactly one intersection per corresponding pair, and Euler's formula gives exactly N independent constraints.
    Used in Sec. V A to prove that any loopless plane graph defines an embedded N-qubit stabilizer state whose measurement statistics match the composite operators in the toric code.
  • standard math In a dual cellulation of R^d, each composite X operator on a p-cell anticommutes only with the corresponding Z operator on the dual (d-p)-cell; all other pairs commute.
    This standard property of dual cellulations is used in Sec. V B to guarantee the Pauli algebra required for the cellulation games, though it is not proved in the paper.
  • standard math The cellulation of R^d includes an unbounded d-cell so that the complex is homotopy equivalent to the d-sphere, with Euler characteristic 1+(-1)^d.
    Stated in Sec. V B 1 before Eq. (23); it is needed so the number of independent stabilizer constraints equals N and the embedded state is uniquely specified.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Braiding for the win: Harnessing braiding statistics in topological states to win quantum games." pith.science (2026). https://pith.science/paper/UJRKT7P5

@misc{pith2026241214288,
  author       = {Pith},
  title        = {Pith review of: Braiding for the win: Harnessing braiding statistics in topological states to win quantum games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UJRKT7P5}},
  note         = {Machine review of arXiv:2412.14288}
}
read the original abstract

Nonlocal quantum games provide proof of principle that quantum resources can confer advantage at certain tasks. They also provide a compelling way to explore the computational utility of phases of matter on quantum hardware. In a recent manuscript [Hart et al., arXiv:2403.04829] we demonstrated that a toric code resource state conferred advantage at a certain nonlocal game, which remained robust to small deformations of the resource state. In this manuscript we demonstrate that this robust advantage is a generic property of resource states drawn from topological or fracton ordered phases of quantum matter. To this end, we illustrate how several other states from paradigmatic topological and fracton ordered phases can function as resources for suitably defined nonlocal games, notably the three-dimensional toric-code phase, the X-cube fracton phase, and the double-semion phase. The key in every case is to design a nonlocal game that harnesses the characteristic braiding processes of a quantum phase as a source of contextuality. We unify the strategies that take advantage of mutual statistics by relating the operators to be measured to order and disorder parameters of an underlying generalized symmetry-breaking phase transition. Finally, we massively generalize the family of games that admit perfect strategies when codewords of homological quantum error-correcting codes are used as resources.

Figures

Figures reproduced from arXiv: 2412.14288 by the authors.

Figure 1
Figure 1. FIG. 1. Lattice-independent illustration of the operator configurations [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) A membrane operator that creates four well-separated fractons (depicted by the shaded blue cubes) at its corners. Bold red edges host [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) An effective nine-qubit state defined by a graph. The blue [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Microscopic process used to extract the exchange statistics [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

70 extracted references · 53 canonical work pages

  1. [16]

    A. K. Daniel and A. Miyake, Quantum computational advantage with string order parameters of one-dimensional symmetry- protected topological order, Phys. Rev. Lett.126, 090505 (2021)

    work page 2021

  2. [1]

    More spatial dimensions 9

    work page

  3. [2]

    Order and disorder parameters 5 IV

    Increasing the number of players 4 C. Order and disorder parameters 5 IV. Other topological and fracton orders 6 A. 3D toric code 6

    work page

  4. [3]

    X-cube model 7 V

    2-form symmetry 6 B. X-cube model 7 V. Beyond the parity game 8 A. Embedding other states in two dimensions 8 B. Homological codes and cellulation games 9

    work page

  5. [4]

    Braiding for the win: Harnessing braiding statistics in topological states to win quantum games

    Cellulation games 10 VI. Beyond mutual statistics: double semion model 10 VII. Discussion 12 References 13 I. INTRODUCTION Measurements on disjoint parts of a quantum state can fa- mously be more correlated than any classical hidden-variable model would permit [ 1]. This result was recast in the lan- guage of nonlocal quantum games by Mermin [2, 3], where...

    work page Pith review arXiv 2024

  6. [5]

    (10) See (12) for the relative position of these operators on the lattice

    Quantum strategy In order to arrive at a perfect quantum mechanical strategy, consider the following loop segments: 𝑋1= , 𝑋 2= , 𝑋 3= , 𝑍1= , 𝑍 2= , 𝑍 3= . (10) See (12) for the relative position of these operators on the lattice. By construction,𝑋𝑖 and𝑍𝑗 only intersect exactly once for𝑖= 𝑗, implying that these operators satisfy the same commutation relat...

    work page

  7. [6]

    𝑋1𝑋2𝑋3∞𝑍2𝑍3= 𝑋1(𝑋2𝑍2)(𝑍3𝑋3),

    Increasing the number of players For games involving𝑃 >3 players [22–24], the operators and the strategy enacted by the players is analogous. Specifically, the operators are shown schematically in Fig. 1 for 𝑃 = 3, 5 and generic𝑃 (see also Ref. [16]). Each player has access to the qubits in the support of two loop segments𝑋𝑖 (on the direct lattice) and𝑍𝑖 ...

    work page

  8. [7]

    All 𝐴𝑒 and 𝐵𝑐 operators share either two or zero faces and hence commute; the ground states of the model have all𝐴𝑒=𝐵𝑐 = 1

    1-form symmetry We work with a 3D version of Kitaev’s toric code [27, 35] with degrees of freedom defined on the elementary faces 𝑓 of an𝐿×𝐿×𝐿 cubic lattice 𝐻3TC=− ∑︁ 𝑒 𝐴𝑒− ∑︁ 𝑐 𝐵𝑐, (14) where𝐴𝑒= Q 𝜕𝑓∋𝑒𝑍𝑓 and𝐵𝑐 = Q 𝑓∈𝜕𝑐𝑋𝑓 are associated to edges𝑒 and cubes (or cells) 𝑐, respectively. All 𝐴𝑒 and 𝐵𝑐 operators share either two or zero faces and hence commute...

    work page

Show all 70 references
  1. [8]

    braiding

    2-form symmetry An entirely analogous construction exists that uses the 2-form symmetry of the model. To make conventions consistent with the previous examples, it is convenient to consider a modified version of (14) in which degrees of freedom are situated on the edges of an𝐿...

  2. [9]

    More spatial dimensions Moving from two to𝑑 spatial dimensions, we now consider a cellulation of R𝑑 (see, e.g., Ref. [46]). Specifically, we consider the cellulation X , which consists of 0-cells (vertices), 1-cells (edges), 2-cells (faces), and 3-cells (volumes), etc., and it...

  3. [10]

    topological games

    Cellulation games Given such a cellulation X , we introduce a game for which codewords of the underlying homological code provide a perfect quantum strategy. Associate players to 𝑝-cells of X , and classical bits x∈{ 0, 1}𝑁 to a fixed basis for𝐵𝑝 and𝐵𝑝 [47]. For example, consi...

  4. [11]

    J. S. Bell, On the problem of hidden variables in quantum mechanics, Rev. Mod. Phys.38, 447 (1966)

    1966

  5. [12]

    N. D. Mermin, Hidden variables and the two theorems of john bell, Rev. Mod. Phys.65, 803 (1993)

    1993

  6. [13]

    N. D. Mermin, Quantum mysteries revisited, Am. J. Phys 58, 731 (1990)

    1990

  7. [14]

    Kochen and E

    S. Kochen and E. Specker, The problem of hidden variables in quantum mechanics, Journal of Mathematics and Mechanics 17 (1967)

    1967

  8. [15]

    Brassard, A

    G. Brassard, A. Broadbent, and A. Tapp, Quantum pseudo- telepathy, Foundations of Physics35, 1877 (2005)

    2005

  9. [17]

    A. K. Daniel, Y. Zhu, C. H. Alderete, V. Buchemmavari, A. M. Green, N. H. Nguyen, T. G. Thurtell, A. Zhao, N. M. Linke, and A. Miyake, Quantum computational advantage attested by nonlocal games with the cyclic cluster state, Phys. Rev. Res.4, 033068 (2022)

    2022

  10. [18]

    Bravyi, D

    S. Bravyi, D. Gosset, R. Koenig, and M. Tomamichel, Quantum advantage with noisy shallow circuits, Nature Physics16, 1040 (2020)

    2020

  11. [19]

    Bravyi, D

    S. Bravyi, D. Gosset, and R. K ¨onig, Quantum advantage with shallow circuits, Science 362, 308 (2018)

    2018

  12. [20]

    Jaffali and F

    H. Jaffali and F. Holweck, Two new non-equivalent three-qubit CHSH games, Quant. Inf. Comput. 24, 0438 (2024)

    2024

  13. [21]

    Sheffer, D

    M. Sheffer, D. Azses, and E. G. Dalla Torre, Playing quantum nonlocal games with six noisy qubits on the cloud, Advanced Quantum Technologies 5, 2100081 (2022)

    2022

  14. [22]

    V. B. Bulchandani, F. J. Burnell, and S. L. Sondhi, A multiplayer multiteam nonlocal game for the toric code, Phys. Rev. B 107, 035409 (2023)

    2023

  15. [23]

    V. B. Bulchandani, F. J. Burnell, and S. L. Sondhi, Playing nonlocal games with phases of quantum matter, Phys. Rev. B 107, 045412 (2023)

    2023

  16. [24]

    C. Lin, V. B. Bulchandani, and S. L. Sondhi, Quantum tasks assisted by quantum noise (2023), arXiv:2308.10969 [quant-ph]

    2023 arXiv

  17. [25]

    Natarajan and C

    A. Natarajan and C. Nirkhe, The status of the quantum pcp conjecture (games version) (2024), arXiv:2403.13084 [quant- ph]

    2024 arXiv

  18. [26]

    O. Hart, D. T. Stephen, D. J. Williamson, M. Foss-Feig, and R. Nandkishore, Playing nonlocal games across a topo- logical phase transition on a quantum computer (2024), arXiv:2403.04829 [quant-ph]

    2024 arXiv

  19. [27]

    G.-Y. Zhu, N. Tantivasadakarn, A. Vishwanath, S. Trebst, and R. Verresen, Nishimori’s cat: Stable long-range entanglement from finite-depth unitaries and weak measurements, Phys. Rev. Lett. 131, 200201 (2023)

    2023

  20. [28]

    J. Y. Lee, W. Ji, Z. Bi, and M. P. A. Fisher, Decoding measurement- prepared quantum phases and transitions: from ising model to gauge theory, and beyond (2022), arXiv:2208.11699 [cond- mat.str-el]

    2022 arXiv

  21. [29]

    E. Chen, G. Zhu, R. Verresen, et al., Nishimori transition across the error threshold for constant-depth quantum circuits, Nature Physics 10.1038/s41567-024-02696-6 (2024)

    2024 doi

  22. [30]

    D. M. Greenberger, M. A. Horne, and A. Zeilinger, Going beyond bell’s theorem, in Bell’s theorem, quantum theory and conceptions of the universe (Springer, 1989) pp. 69–72

    1989

  23. [31]

    D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, Bell’s theorem without inequalities, American Journal of Physics 58, 1131 (1990)

    1990

  24. [32]

    N. D. Mermin, Extreme quantum entanglement in a superposition of macroscopically distinct states, Phys. Rev. Lett. 65, 1838 (1990)

    1990

  25. [33]

    Brassard, A

    G. Brassard, A. Broadbent, and A. Tapp, Multi-party pseudo- telepathy, inAlgorithms and Data Structures, edited by F. Dehne, J.-R. Sack, and M. Smid (Springer Berlin Heidelberg, Berlin, Heidelberg, 2003) pp. 1–11

    2003

  26. [34]

    Brassard, A

    G. Brassard, A. Broadbent, and A. Tapp, Recasting mermin’s multi-player game into the framework of pseudo-telepathy, Quan- tum Information and Computation 5, 538 (2005)

    2005

  27. [35]

    Anders and D

    J. Anders and D. E. Browne, Computational power of correlations, Phys. Rev. Lett.102, 050502 (2009)

    2009

  28. [36]

    A. C.-C. Yao, Probabilistic computations: Toward a unified mea- sure of complexity, in 18th Annual Symposium on Foundations of Computer Science (sfcs 1977) (1977) pp. 222–227

    1977

  29. [37]

    Kitaev, Fault-tolerant quantum computation by anyons, Ann

    A. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303, 2 (2003)

    2003

  30. [38]

    Haah, An invariant of topologically ordered states under local unitary transformations, Communications in Mathematical Physics 342, 771 (2016)

    J. Haah, An invariant of topologically ordered states under local unitary transformations, Communications in Mathematical Physics 342, 771 (2016)

    2016

  31. [39]

    Fradkin, Disorder operators and their descendants, Journal of Statistical Physics 167, 427 (2017)

    E. Fradkin, Disorder operators and their descendants, Journal of Statistical Physics 167, 427 (2017)

    2017

  32. [40]

    A. Dua, I. H. Kim, M. Cheng, and D. J. Williamson, Sorting topological stabilizer models in three dimensions, Phys. Rev. B 100, 155137 (2019)

    2019

  33. [41]

    Nussinov and G

    Z. Nussinov and G. Ortiz, A symmetry principle for topological quantum order, Annals of Physics 324, 977 (2009)

    2009

  34. [42]

    Gaiotto, A

    D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Generalized global symmetries, J. High Energ. Phys. 2015 (172)

    2015

  35. [43]

    McGreevy, Generalized symmetries in condensed matter, Annual Review of Condensed Matter Physics 14, 57 (2023)

    J. McGreevy, Generalized symmetries in condensed matter, Annual Review of Condensed Matter Physics 14, 57 (2023)

    2023

  36. [44]

    Fredenhagen and M

    K. Fredenhagen and M. Marcu, Charged states in Z2 gauge theo- ries, Communications in Mathematical Physics 92, 81 (1983)

    1983

  37. [45]

    Hamma, P

    A. Hamma, P. Zanardi, and X.-G. Wen, String and membrane condensation on three-dimensional lattices, Phys. Rev. B 72, 035307 (2005)

    2005

  38. [46]

    Expressions such as𝑓∈𝜕𝑅∗ should be understood as 𝑓∈∗𝜕𝑅∗

    We will often omit the isomorphism∗ between a cell complex𝑋 and its dual𝑋∗ for brevity. Expressions such as𝑓∈𝜕𝑅∗ should be understood as 𝑓∈∗𝜕𝑅∗

  39. [47]

    R. M. Nandkishore and M. Hermele, Fractons, Annual Review of Condensed Matter Physics 10, 295 (2019)

    2019

  40. [48]

    Castelnovo, C

    C. Castelnovo, C. Chamon, and D. Sherrington, Quantum me- chanical and information theoretic view on classical glass transi- 13 BRAIDING FOR THE WIN: HARNESSING BRAIDING ... tions, Phys. Rev. B81, 184303 (2010)

    2010

  41. [49]

    Vijay, J

    S. Vijay, J. Haah, and L. Fu, Fracton topological order, general- ized lattice gauge theory, and duality, Phys. Rev. B94, 235157 (2016)

    2016

  42. [50]

    H. Ma, E. Lake, X. Chen, and M. Hermele, Fracton topological order via coupled layers, Phys. Rev. B95, 245126 (2017)

    2017

  43. [51]

    A self-loop corresponds to a trivial, disentangled qubit, since either𝑋𝑖 or𝑍𝑖 belongs to the stabilizer group for some𝑖 corresponding to the self-loop

    We consider graphs𝐺 where𝐺 and its geometric dual𝐺∗ are loopless, i.e., they contain no self-loops. A self-loop corresponds to a trivial, disentangled qubit, since either𝑋𝑖 or𝑍𝑖 belongs to the stabilizer group for some𝑖 corresponding to the self-loop

  44. [52]

    Dennis, A

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, Journal of Mathematical Physics 43, 4452 (2002)

    2002

  45. [53]

    [12], the strategy presented herein does not require the application of a nonlocal unitary operator, but gives rise to the same𝑝q(|𝜓⟩) given some generic resource state|𝜓⟩

    Unlike Ref. [12], the strategy presented herein does not require the application of a nonlocal unitary operator, but gives rise to the same𝑝q(|𝜓⟩) given some generic resource state|𝜓⟩

  46. [54]

    M. H. Freedman and D. A. Meyer, Projective plane and planar quantum codes, Foundations of Computational Mathematics 1, 325 (2001)

    2001

  47. [55]

    Bombin and M

    H. Bombin and M. A. Martin-Delgado, Homological error cor- rection: Classical and quantum codes, Journal of Mathematical Physics 48, 052105 (2007)

    2007

  48. [56]

    N. P. Breuckmann,Homological quantum codes beyond the toric code, Dissertation, RWTH Aachen University, Aachen (2017)

    2017

  49. [57]

    If additional stabilizer generators are included, the map from x to bits a and b is surjective but not injective

    The requirement that the bits are associated to a linearly indepen- dent set of stabilizer generators can straightforwardly be relaxed. If additional stabilizer generators are included, the map from x to bits a and b is surjective but not injective

  50. [58]

    M. A. Levin and X.-G. Wen, String-net condensation: A physical mechanism for topological phases, Phys. Rev. B 71, 045110 (2005)

    2005

  51. [59]

    P. K. Aravind, Bell’s theorem without inequalities and only two distant observers, Foundations of Physics Letters 15, 397 (2002)

    2002

  52. [60]

    P. K. Aravind, A simple demonstration of bell’s theorem involv- ing two observers and no probabilities or inequalities (2003), arXiv:quant-ph/0206070 [quant-ph]

    2003 arXiv

  53. [61]

    N. D. Mermin, Simple unified form for the major no-hidden- variables theorems, Phys. Rev. Lett.65, 3373 (1990)

    1990

  54. [62]

    T. D. Ellison, Y.-A. Chen, A. Dua, W. Shirley, N. Tantivasadakarn, and D. J. Williamson, Pauli stabilizer models of twisted quantum doubles, PRX Quantum 3, 010353 (2022)

    2022

  55. [63]

    Kawagoe and M

    K. Kawagoe and M. Levin, Microscopic definitions of anyon data, Phys. Rev. B101, 115113 (2020)

    2020

  56. [64]

    Laversanne-Finot, A

    A. Laversanne-Finot, A. Ketterer, M. R. Barros, S. P. Walborn, T. Coudreau, A. Keller, and P. Milman, General conditions for maximal violation of non-contextuality in discrete and continuous variables, Journal of Physics A: Mathematical and Theoretical 50, 155304 (2017)

    2017

  57. [65]

    Alternatively, phase estimation can be performed, which is exact in the present setting [59]

    Measurement of the unitary operators 𝑈𝑖 = P 𝑗𝜔𝑠 𝑗|𝑗⟩⟨ 𝑗| with 𝑠𝑗∈ Z𝑑 may be thought of as measuring the Hermitian operatorP 𝑗𝑠𝑗|𝑗⟩⟨ 𝑗|. Alternatively, phase estimation can be performed, which is exact in the present setting [59]

  58. [66]

    Devakul and D

    T. Devakul and D. J. Williamson, Fractalizing quantum codes, Quantum 5, 438 (2021)

    2021

  59. [67]

    H. Song, N. Tantivasadakarn, W. Shirley, and M. Hermele, Fracton self-statistics, Phys. Rev. Lett.132, 016604 (2024)

    2024

  60. [68]

    Gidney, Stability Experiments: The Overlooked Dual of Memory Experiments, Quantum 6, 786 (2022)

    C. Gidney, Stability Experiments: The Overlooked Dual of Memory Experiments, Quantum 6, 786 (2022)

    2022

  61. [69]

    C. Okay, S. Roberts, S. D. Bartlett, and R. Raussendorf, Topo- logical proofs of contextuality in qunatum mechanics, Quantum Information & Computation 17, 1135 (2017)

    2017

  62. [70]

    Delcamp and A

    C. Delcamp and A. Tiwari, On 2-form gauge models of topolog- ical phases, Journal of High Energy Physics 2019, 1 (2019). 14

    2019

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.