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Pauli stabilizer formalism for topological quantum field theories and generalized statistics

T0 review · 2 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper constructs Pauli stabilizer models whose loop excitations are fermionic — a 24-step lattice process picks up phase −1 — and uses this to realize all (4+1)D Dijkgraaf–Witten 2-form gauge theories.

desk verdict A serious construction paper with a real gap in the Pauli completeness proof; the central fermionic-loop example likely holds, but the general claims need a rigorous locality-to-boundary argument. read the letter →

arxiv 2601.00064 v3 pith:DRLLDVNJ submitted 2025-12-31 quant-ph cond-mat.str-elhep-thmath.QA

Yitao Feng , Hanyu Xue , Ryohei Kobayashi , Po-Shen Hsin , Yu-An Chen This is my paper
classification quant-phcond-mat.str-elhep-thmath.QA
keywords Paulistabilizercodestopologicalquantumfieldtheoryhigher-formgaugefermionicloopstatisticsDijkgraaf-Wittencondensationhighercupproductsextendedexcitations
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

The paper aims to show that a wide class of higher-dimensional topological quantum field theories can be realized exactly as Pauli stabilizer Hamiltonians, with extended excitations whose generalized statistics can be computed directly on the lattice. Its central example is the (4+1)D fermionic-loop toric code, obtained by condensing the e²m² loop in the Z₄ loop-only toric code; the paper proves that the resulting flux loop has fermionic statistics, meaning a 24-step loop-flipping process produces a phase of −1. More broadly, it constructs Pauli stabilizer models for all (4+1)D Dijkgraaf–Witten 2-form gauge theories classified by H⁵(B²G,U(1)), and extends the construction to arbitrary dimensions for the fermionic-loop family, as well as to membrane and volume toric codes. A sympathetic reader would care because these are explicit commuting-projector Hamiltonians whose excitations, fusion rules, and statistical phases are microscopically computable, which is valuable both for understanding higher-dimensional topological order and for designing quantum codes.

What carries the argument

The load-bearing construction is loop condensation: starting from a Z₄ toric code with qudits on 2-cells, the Hamiltonian adds hopping terms C_f that condense the e²m² loop and keeps only those stabilizers commuting with them. The decorated flux-membrane operator Ṽ^F_λ, built from X operators with higher cup-product corrections, creates the flux-loop excitation, and its commutator algebra is evaluated using higher cup products ∪_i, which are lattice cohomology operations generalizing the ordinary cup product. The statistical invariant is the 24-step loop-flipping process µ₂₄ = [U₀₁₂,U₀₃₄]²[U₀₁₃,U₀₂₄]²[U₀₁₄,U₀₂₃]²; the proof reduces it to an integral involving p₀∪₄p₀ = p₀ (mod 2), giving −1.

What would settle it

A direct numerical evaluation of the 24-step loop-flipping phase µ₂₄ on a small (4+1)D periodic lattice that returns +1 would falsify Theorem 2.1. A complementary check is to search for a local Pauli operator that commutes with every stabilizer of the condensed Hamiltonian but is not a product of stabilizers; its existence would show the model has extra excitations beyond the claimed TQFT.

Watch

Extended reading notes

Core claim

The central claim is that condensing the e²m² loop in the (4+1)D Z₄ loop-only toric code produces a stabilizer Hamiltonian whose ground state realizes the twisted Z₂ 2-form gauge theory with action ¼ b₂∪δb₂, and whose flux-loop excitation carries fermionic loop statistics: the 24-step loop-flipping unitary process yields the phase µ₂₄ = −1 (Theorem 2.1). The same condensation mechanism is shown to realize every (4+1)D Dijkgraaf–Witten 2-form gauge theory classified by H⁵(B²G,U(1)), and, using higher cup products, the fermionic-loop toric codes extend to arbitrary spatial dimension d≥4. The paper also formulates a Pauli-based framework for generalized statistics, constructing unitary processe

Load-bearing premise

The construction depends on the unproved 'Pauli completeness' of the condensed models: that every local Pauli operator commuting with the condensed stabilizers is itself a product of those stabilizers, with the locality-to-cohomology step in Sec. 3.3 asserted rather than demonstrated.

Editorial extensions

If this is right

  • If the central claim is correct, the (4+1)D fermionic-loop toric code is an explicit Pauli stabilizer realization of the w₃ (equivalently Sq²Sq¹) twisted Z₂ 2-form gauge theory.
  • All twisted (4+1)D Dijkgraaf–Witten 2-form gauge theories classified by H⁵(B²G,U(1)) admit lattice stabilizer Hamiltonians obtained by systematic loop condensation from stacks of Z₄ toric codes.
  • The fermionic-loop construction extends to every spatial dimension d≥4, yielding a family of Z₂ topological orders with fermionic flux loops, bosonic (d−3)-brane charges, and mutual −1 braiding between them.
  • The Pauli-based statistics framework gives computable unitary tests for generalized statistics, including the semionic membrane in (6+1)D, fermionic membranes, fermionic volumes, and anyonic p-brane statistics only for even p.
  • These explicit stabilizer models enlarge the design space for higher-dimensional quantum error-correcting codes and self-correcting quantum memories, since their excitation structures and logical operations become microscopically tractable.

Reading between the lines

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

  • The group-extension strategy — lift the target Z₂ theory to a Z₄ gauge theory, condense the appropriate power loop, and let the surviving phase appear as statistics — is a general recipe the paper leaves implicit; it could plausibly be applied to other cohomology classes beyond H⁵(B²G,U(1)).
  • The 'Pauli completeness' step is the point where the realization could silently acquire extra superselection sectors; verifying completeness numerically on small lattices, or finding a counterexample, would directly test whether the condensed model is exactly the claimed TQFT.
  • The paper's conjecture that Pauli statistics embeds into non-Pauli statistics suggests a concrete classification question — which cohomology classes admit quadratic cochain representatives — and a natural test would be to search for a Pauli realization of the non-Pauli Pontryagin Z₃ membrane statistics.
  • The observed periodicity (semion ↔ semionic membrane, fermionic loop ↔ fermionic volume) predicts stabilizer codes in dimensions differing by four; constructing the (10+1)D analogue and checking its statistics would test whether the pattern is robust.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper develops a Pauli stabilizer formalism for higher-form gauge theories and generalized statistics. Its central claims are (i) a (4+1)D fermionic-loop toric code obtained by condensing the e^2 m^2 loop in the Z_4 loop-only toric code, whose flux loop has fermionic loop statistics with 24-step phase -1; (ii) Pauli stabilizer realizations of all (4+1)D Dijkgraaf-Witten 2-form gauge theories classified by H^5(B^2G,U(1)); and (iii) families of higher-dimensional models (fermionic-loop, semionic-membrane, fermionic-membrane, fermionic-volume toric codes) with computable generalized statistics expressed through higher cup products. The paper also proposes a Pauli-adapted classification T_P^*(m) of statistics of extended excitations.

Significance. If the constructions are fully established, this is a substantial advance: it would give explicit commuting-Pauli Hamiltonians for a wide class of higher-form TQFTs, concrete microscopic detectors for loop, membrane, and volume statistics, and the first stabilizer realization of the fermionic-loop order. The algebraic machinery of higher cup products is used in a detailed and largely coherent way; the 24-step calculation in Sec. 2.3 is a worked, hand-verifiable derivation. The paper is also careful to identify what is conjectural (e.g., injectivity of T_P^* to non-Pauli statistics) and what is computational (small cases in Sec. 5.1). However, the proof that the condensed stabilizer models are maximally commutative contains a genuinely load-bearing gap, and this gap propagates to the higher-dimensional families.

major comments (2)
  1. [Sec. 3.3, Eqs. (87)-(89)] The completeness proof is not complete. After deriving A ∈ C_2(M,K⊥) with ∂A ∈ C_1(M,\bar K), the text states: 'Given that P is a local operator, we further derive that A is the boundary of some K⊥/\bar K-valued chain.' This locality-to-cohomology step is asserted, not proved. A finite-support 2-chain whose boundary lies in \bar K need not be a boundary in K⊥/\bar K unless the relevant compactly supported homology vanishes; the hypersquare lattice or non-degeneracy of the cup product does not by itself imply this. The subsequent decomposition A = A_W + ∂A_G and Eq. (89) depend entirely on this step. Since maximal commutativity is the bridge between the condensed Pauli Hamiltonian and the claimed TQFT ground-state sector, the realization of all H^5(B^2G,U(1)) theories and the phase identification of the fermionic-loop example are not yet established. The same unproved completeness logic i
  2. [Sec. 4.1, Eqs. (103)-(107), and Secs. 4.2, 7.2, 8.2] For d ≥ 5, the construction changes character: e and m excitations have different dimensions, so the e^2m^2 condensation of Sec. 2 is replaced by a gauging of an SPT whose cocycle is stated to be trivial. The equivalence of the two forms of the Z_4 toric code Hamiltonian is delegated to an FDQC (Refs. [51,62]) and not demonstrated. More importantly, the condensed Hamiltonian is asserted to have the claimed ground-state sector without proving that its stabilizer group is the full commutant of the added condensation terms. Thus even if the Sec. 3.3 lemma is repaired for (4+1)D, the higher-dimensional families require the same completeness statement, which is not supplied. Please either provide the analogue of the completeness argument for each dimension or state precisely what is being assumed.
minor comments (5)
  1. [Sec. 3.3, text before Eq. (82)] Typo: 'No,w suppose' should be 'Now, suppose'.
  2. [Sec. 5.1, Definition 2 and Eq. (129)] The codomain of φ in Definition 2 is written as A, but φ is a phase, so it must take values in R/2πZ (or R/Z). Please fix this notation.
  3. [Sec. 7.2, Eq. (205)] The Hamiltonian H_{fermionic−particle} in Eq. (205) actually describes the fermionic-membrane toric code and should be renamed accordingly.
  4. [Sec. 3.3, paragraph after Eq. (91)] The statement that K=\bar K follows from |K|·|K⊥|=|G| is too terse: the order equality needs to be combined with the nondegeneracy of the symplectic pairing to conclude subgroup equality. This is standard, but it should be said explicitly.
  5. [Sec. 5.1, computer computations] The claims that 'for small p,d,G we compute T_P directly using a computer' are central to the conjecture T_P^* ⊂ H^{d+2}(B^{d-p}G,R/Z). No code, input data, or reproducible verification is provided. For a quantitative claim of this type, please include at least the computed groups as an ancillary file or table.

Circularity Check

0 steps flagged · score 2.0 of 10

No construction-level circularity: the central 24-step evaluation is a self-contained computation; the main caveat is an unproved locality-to-boundary completeness step, which is a correctness gap, not a reduction by construction.

full rationale

The central derivation chain is not circular. The fermionic-loop toric code is defined by a concrete stabilizer Hamiltonian (Eqs. (14)/(19)) and the claim that its flux loop has mu_24 = -1 is proved in Theorem 2.1 by direct commutator algebra (Eqs. (33)-(39)), not by assuming the conclusion or by fitting any parameter. The '24-step process' is a benchmark/detector imported from prior work (Refs. [41,43]) and is not itself the object being derived. Similarly, the general Pauli-statistics framework in Sec. 5 is definitional; the statistical expressions such as Eq. (149) are shown nontrivial by constructing explicit realizations (Eq. (151)), and the non-Pauli comparison in Appendix B is presented as an embedding with an explicitly open injectivity question. The self-citations to Refs. [41-44,57] supply definitions, uniqueness statements, and prior techniques, but none of these citations substitutes for the proof of the new models' statistics. The only significant caveat is Sec. 3.3: the assertion 'Given that P is a local operator, we further derive that A is the boundary of some K^perp/bar-K-valued chain' is an unproved locality-to-cohomology step on which the Pauli completeness of the condensed models rests. This is a genuine mathematical gap (a compact-support homology statement is needed and not supplied), and it is load-bearing for the claim that the condensed Hamiltonians realize all H^5(B^2G,U(1)) theories. However, a missing lemma is not circular: the conclusion is not equivalent to an input by construction. The models, actions, and statistics are independently specified, and no fitted parameter forces the predicted phase. Accordingly the circularity score is low.

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

No fitted free parameters appear; all couplings are fixed by the Z_N qudit algebra and the chosen cocycle data. The key unproved inputs are algebraic-topology identities, the condensation/completeness assumptions, and the computer-based classification that is not reproduced. The new objects are models and phases with explicit lattice operators rather than unexplained new physical entities.

assumptions (4)
  • standard math Higher cup products ∪_i on hypercubic lattices satisfy the Leibniz/coboundary identities and vanishing/support properties used throughout (e.g., Eqs. (24), (36), (122), (202), (247)).
    Invoked in every statistics proof; imported from Refs. [50-52] and Steenrod's higher cup products; not reproved in this paper.
  • domain assumption The condensation procedure—adding a hopping term C_f and retaining the commuting stabilizer subgroup—yields a commuting-projector Hamiltonian whose ground-state sector and topological order equal the field-theoretic condensation result.
    Used to identify H_condensed with the twisted 2-form gauge theory in Secs. 2.2 and 3.2; standard in this literature but not independently justified here.
  • domain assumption Statistics of extended excitations can be extracted from the Pauli phase function φ(s',s) with axioms (129), (133), (135), and the 24-step/loop-flipping process reduces to Eq. (31) in the Pauli setting.
    Core to the Sec. 5 framework; adapted from Refs. [43,44]. The paper states the non-Pauli definitions agree, but the Pauli reduction is its own formalism.
  • ad hoc to paper The computer calculations underlying T_P(m_p(...)) for small p,d,G are correct, and the dimension-reduction identity ∫_{M^d}=∫_{∂∆^7} in Sec. 7.2 is valid.
    No code or full proof is supplied; the reduction is illustrated only with one cup-1 example. These are load-bearing for the claimed T_P classifications and fermionic-membrane statistics.

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Cite this review

Pith. "Pith review of Pauli stabilizer formalism for topological quantum field theories and generalized statistics." pith.science (2026). https://pith.science/paper/DRLLDVNJ

@misc{pith2026260100064,
  author       = {Pith},
  title        = {Pith review of: Pauli stabilizer formalism for topological quantum field theories and generalized statistics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DRLLDVNJ}},
  note         = {Machine review of arXiv:2601.00064}
}
abstract

Topological quantum field theory (TQFT) provides a unifying framework for describing topological phases of matter and for constructing quantum error-correcting codes, playing a central role across high-energy physics, condensed matter, and quantum information. A central challenge is to formulate topological order on lattices and to extract the properties of topological excitations from microscopic Hamiltonians. In this work, we construct new classes of lattice gauge theories as Pauli stabilizer models, realizing a wide range of TQFTs in general dimensions. We develop a lattice description of extended excitations and systematically determine their generalized statistics. Our main example is the (4+1)D fermionic-loop toric code, obtained by condensing the $e^2m^2$-loop in the (4+1)D $\mathbb Z_4$ toric code. We show that the loop excitation exhibits fermionic loop statistics: the 24-step loop-flipping process yields a phase of $-1$. Our Pauli stabilizer models realize all twisted 2-form gauge theories in (4+1)D, the higher-form Dijkgraaf-Witten TQFT classified by $H^5(B^2G,U(1))$. Beyond (4+1)D, the fermionic-loop toric codes form a family of $\mathbb Z_2$ topological orders in arbitrary dimensions, realized as explicit Pauli stabilizer codes using $\mathbb Z_4$ qudits. Finally, we develop a Pauli-based framework that defines generalized statistics for extended excitations in any dimension, yielding computable lattice unitary processes to detect nontrivial statistics. For example, we propose anyonic membrane statistics in (6+1)D, as well as fermionic membrane and volume statistics in arbitrary dimensions. We construct new families of $\mathbb Z_2$ topological orders: the fermionic-membrane toric code and the fermionic-volume toric code. In addition, we demonstrate that $p$-dimensional excitations in $2p+2$ spatial dimensions can support anyonic $p$-brane statistics for only even $p$.

Figures

Figures reproduced from arXiv: 2601.00064 by the authors.

Figure 1
Figure 1. Simplicial complexes used in different spacetime dimensions (adapted from Ref. [ [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. The 24-step process for detecting the statistics of loop excitations with [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

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Reference graph

Works this paper leans on

93 extracted references · 12 linked inside Pith · cited by 5 Pith papers

  1. [44]

    Statistics of Abelian topological excitations,

    H. Xue, “Statistics of Abelian topological excitations,”arXiv:2412.07653 [quant-ph]

    arXiv

  2. [1]

    Topological Gauge Theories and Group Cohomology,

    R. Dijkgraaf and E. Witten, “Topological Gauge Theories and Group Cohomology,”Commun. Math. Phys.129(1990) 393

    1990

  3. [2]

    Topological quantum field theories,

    M. Atiyah, “Topological quantum field theories,”Publications Math´ ematiques de l’Institut des Hautes ´Etudes Scientifiques68no. 1, (1988) 175–186

    1988

  4. [3]

    Topological quantum memory,

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory,”Journal of Mathematical Physics43no. 9, (2002) 4452–4505

    2002

  5. [4]

    Topological quantum computation,

    M. H. Freedman, A. Kitaev, M. J. Larsen, and Z. Wang, “Topological quantum computation,” Bulletin of the American Mathematical Society40no. 1, (2003) 31–38. 43

    2003

  6. [5]

    Non-abelian anyons and topological quantum computation,

    C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-abelian anyons and topological quantum computation,”Rev. Mod. Phys.80(2008) 1083–1159

    2008

  7. [6]

    Symmetry protected topological orders and the group cohomology of their symmetry group,

    X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, “Symmetry protected topological orders and the group cohomology of their symmetry group,”Phys. Rev. B87(2013) 155114

    2013

  8. [7]

    Fault-tolerant quantum computation by anyons,

    A. Kitaev, “Fault-tolerant quantum computation by anyons,”Annals of Physics303no. 1, (2003) 2–30

    2003

Show all 93 references
  1. [8]

    String-net condensation: A physical mechanism for topological phases,

    M. A. Levin and X.-G. Wen, “String-net condensation: A physical mechanism for topological phases,” Phys. Rev. B71(2005) 045110

    2005

  2. [9]

    Twisted quantum double model of topological phases in two dimensions,

    Y. Hu, Y. Wan, and Y.-S. Wu, “Twisted quantum double model of topological phases in two dimensions,”Phys. Rev. B87(2013) 125114

    2013

  3. [10]

    Anyon condensation and tensor categories,

    L. Kong, “Anyon condensation and tensor categories,”Nuclear Physics B886(2014) 436–482

    2014

  4. [11]

    Braiding statistics approach to symmetry-protected topological phases,

    M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases,” Phys. Rev. B86(2012) 115109

    2012

  5. [12]

    Pauli stabilizer models of twisted quantum doubles,

    T. D. Ellison, Y.-A. Chen, A. Dua, W. Shirley, N. Tantivasadakarn, and D. J. Williamson, “Pauli stabilizer models of twisted quantum doubles,”PRX Quantum3(2022) 010353

    2022

  6. [13]

    Fusion rules and shrinking rules of topological orders in five dimensions,

    Y. Huang, Z.-F. Zhang, and P. Ye, “Fusion rules and shrinking rules of topological orders in five dimensions,”Journal of High Energy Physics2023no. 11, (2023) 210

    2023

  7. [14]

    Diagrammatics, pentagon equations, and hexagon equations of topological orders with loop- and membrane-like excitations,

    Y. Huang, Z.-F. Zhang, and P. Ye, “Diagrammatics, pentagon equations, and hexagon equations of topological orders with loop- and membrane-like excitations,”Journal of High Energy Physics2025 no. 6, (2025) 238

    2025

  8. [15]

    Gottesman,Stabilizer Codes and Quantum Error Correction

    D. Gottesman,Stabilizer Codes and Quantum Error Correction. PhD thesis, California Institute of Technology, 1997.arXiv:quant-ph/9705052

    1997 arXiv

  9. [16]

    Quantum error correction for quantum memories,

    B. M. Terhal, “Quantum error correction for quantum memories,”Reviews of Modern Physics87 no. 2, (2015) 307–346

    2015

  10. [17]

    Topological phases and quantum computation,

    A. Kitaev and C. Laumann, “Topological phases and quantum computation,”arXiv:0904.2771 [cond-mat.mes-hall]

    arXiv

  11. [18]

    Self-correcting quantum computers,

    H. Bombin, R. W. Chhajlany, M. Horodecki, and M. A. Martin-Delgado, “Self-correcting quantum computers,”New Journal of Physics15no. 5, (2013) 055023

    2013

  12. [19]

    Non-abelian self-correcting quantum memory and transversal non-clifford gate beyond then 1/3 distance barrier,

    P.-S. Hsin, R. Kobayashi, and G. Zhu, “Non-abelian self-correcting quantum memory and transversal non-clifford gate beyond then 1/3 distance barrier,”PRX Quantum6(2025) 040360

    2025

  13. [20]

    Topological quantum distillation,

    H. Bombin and M. A. Martin-Delgado, “Topological quantum distillation,”Phys. Rev. Lett.97(2006) 180501

    2006

  14. [21]

    Unfolding the color code,

    A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code,”New Journal of Physics17 no. 8, (2015) 083026

    2015

  15. [22]

    Quantum kronecker sum-product low-density parity-check codes with finite rate,

    A. A. Kovalev and L. P. Pryadko, “Quantum kronecker sum-product low-density parity-check codes with finite rate,”Phys. Rev. A88(2013) 012311

    2013

  16. [23]

    Distance bounds for generalized bicycle codes,

    R. Wang and L. P. Pryadko, “Distance bounds for generalized bicycle codes,”Symmetry14no. 7, (2022)

    2022

  17. [24]

    High-threshold and low-overhead fault-tolerant quantum memory,

    S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, “High-threshold and low-overhead fault-tolerant quantum memory,”Nature627no. 8005, (2024) 778–782

    2024

  18. [25]

    Coprime Bivariate Bicycle Codes and Their Layouts on Cold Atoms,

    M. Wang and F. Mueller, “Coprime Bivariate Bicycle Codes and Their Layouts on Cold Atoms,” arXiv:2408.10001 [quant-ph]. 44

  19. [26]

    Rate adjustable bivariate bicycle codes for quantum error correction,

    M. Wang and F. Mueller, “Rate adjustable bivariate bicycle codes for quantum error correction,” in 2024 IEEE International Conference on Quantum Computing and Engineering (QCE), vol. 02, pp. 412–413. 2024

    2024

  20. [27]

    Low-Overhead Entangling Gates From Generalised Dehn Twists,

    R. Tiew and N. P. Breuckmann, “Low-Overhead Entangling Gates From Generalised Dehn Twists,” IEEE Trans. Info. Theor.71no. 7, (2025) 5452–5468

    2025

  21. [28]

    Ambiguity Clustering: an accurate and efficient decoder for qLDPC codes,

    S. Wolanski and B. Barber, “Ambiguity Clustering: an accurate and efficient decoder for qLDPC codes,”arXiv:2406.14527 [quant-ph]

    arXiv

  22. [29]

    Lowering connectivity requirements for bivariate bicycle codes using morphing circuits,

    M. H. Shaw and B. M. Terhal, “Lowering connectivity requirements for bivariate bicycle codes using morphing circuits,”Phys. Rev. Lett.134(2025) 090602

    2025

  23. [30]

    Multivariate bicycle codes,

    L. Voss, S. J. Xian, T. Haug, and K. Bharti, “Multivariate bicycle codes,”Phys. Rev. A111(2025) L060401

    2025

  24. [31]

    Toward a 2D local implementation of quantum low-density parity-check codes,

    N. Berthusen, D. Devulapalli, E. Schoute, A. M. Childs, M. J. Gullans, A. V. Gorshkov, and D. Gottesman, “Toward a 2D local implementation of quantum low-density parity-check codes,”PRX Quantum6(2025) 010306

    2025

  25. [32]

    Logical operators and fold-transversal gates of bivariate bicycle codes,

    J. N. Eberhardt and V. Steffan, “Logical operators and fold-transversal gates of bivariate bicycle codes,”IEEE Transactions on Information Theory71no. 2, (2025) 1140–1152

    2025

  26. [33]

    Generalized toric codes on twisted tori for quantum error correction,

    Z. Liang, K. Liu, H. Song, and Y.-A. Chen, “Generalized toric codes on twisted tori for quantum error correction,”PRX Quantum6(2025) 020357

    2025

  27. [34]

    Self-dual bivariate bicycle codes with transversal Clifford gates,

    Z. Liang and Y.-A. Chen, “Self-dual bivariate bicycle codes with transversal Clifford gates,” arXiv:2510.05211 [quant-ph]

  28. [35]

    Structure of 2D topological stabilizer codes,

    H. Bomb ´ ın, “Structure of 2D topological stabilizer codes,”Communications in Mathematical Physics 327no. 2, (2014) 387–432

    2014

  29. [36]

    Commuting pauli hamiltonians as maps between free modules,

    J. Haah, “Commuting pauli hamiltonians as maps between free modules,”Communications in Mathematical Physics324no. 2, (2013) 351–399

    2013

  30. [37]

    Algebraic methods for quantum codes on lattices,

    J. Haah, “Algebraic methods for quantum codes on lattices,”Revista colombiana de matematicas50 no. 2, (2016) 299–349

    2016

  31. [38]

    Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices,

    J. Haah, “Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices,”Journal of Mathematical Physics62no. 1, (2021) 012201

    2021

  32. [39]

    Equivalence between fermion-to-qubit mappings in two spatial dimensions,

    Y.-A. Chen and Y. Xu, “Equivalence between fermion-to-qubit mappings in two spatial dimensions,” PRX Quantum4(2023) 010326

    2023

  33. [40]

    Homological invariants of pauli stabilizer codes,

    B. Ruba and B. Yang, “Homological invariants of pauli stabilizer codes,”Communications in Mathematical Physics405no. 5, (2024) 126

    2024

  34. [41]

    Gravitational anomaly of (3 + 1)-dimensionalZ 2 toric code with fermionic charges and fermionic loop self-statistics,

    L. Fidkowski, J. Haah, and M. B. Hastings, “Gravitational anomaly of (3 + 1)-dimensionalZ 2 toric code with fermionic charges and fermionic loop self-statistics,”Phys. Rev. B106(2022) 165135

    2022

  35. [42]

    Exactly solvable lattice Hamiltonians and gravitational anomalies,

    Y.-A. Chen and P.-S. Hsin, “Exactly solvable lattice Hamiltonians and gravitational anomalies,” SciPost Phys.14(2023) 089

    2023

  36. [43]

    Generalized Statistics on Lattices,

    R. Kobayashi, Y. Li, H. Xue, P.-S. Hsin, and Y.-A. Chen, “Generalized Statistics on Lattices,” arXiv:2412.01886 [quant-ph]

  37. [45]

    Loops in 4+1d topological phases,

    X. Chen, A. Dua, P.-S. Hsin, C.-M. Jian, W. Shirley, and C. Xu, “Loops in 4+1d topological phases,” SciPost Phys.15(2023) 001. 45

    2023

  38. [46]

    Quantum ether: Photons and electrons from a rotor model,

    M. Levin and X.-G. Wen, “Quantum ether: Photons and electrons from a rotor model,”Phys. Rev. B 73(2006) 035122

    2006

  39. [47]

    Bosonization in three spatial dimensions and a 2-form gauge theory,

    Y.-A. Chen and A. Kapustin, “Bosonization in three spatial dimensions and a 2-form gauge theory,” Phys. Rev. B100(2019) 245127

    2019

  40. [48]

    Topological invariants for symmetry-protected topological phase entanglers,

    C. Zhang, “Topological invariants for symmetry-protected topological phase entanglers,”Phys. Rev. B 107(2023) 235104

    2023

  41. [49]

    Fermions, strings, and gauge fields in lattice spin models,

    M. Levin and X.-G. Wen, “Fermions, strings, and gauge fields in lattice spin models,”Phys. Rev. B67 (2003) 245316

    2003

  42. [50]

    Higher cup products on hypercubic lattices: Application to lattice models of topological phases,

    Y.-A. Chen and S. Tata, “Higher cup products on hypercubic lattices: Application to lattice models of topological phases,”Journal of Mathematical Physics64no. 9, (2023) 091902

    2023

  43. [51]

    Clifford quantum cellular automata from topological quantum field theories and invertible subalgebras,

    M. Sun, B. Yang, Z. Wang, N. Tantivasadakarn, and Y.-A. Chen, “Clifford quantum cellular automata from topological quantum field theories and invertible subalgebras,”arXiv:2509.07099 [math.QA]

    arXiv

  44. [52]

    Disentangling supercohomology symmetry-protected topological phases in three spatial dimensions,

    Y.-A. Chen, T. D. Ellison, and N. Tantivasadakarn, “Disentangling supercohomology symmetry-protected topological phases in three spatial dimensions,”Phys. Rev. Research3(2021) 013056

    2021

  45. [53]

    Anyons in an exactly solved model and beyond,

    A. Kitaev, “Anyons in an exactly solved model and beyond,”Annals Phys.321no. 1, (2006) 2–111

    2006

  46. [54]

    Products of cocycles and extensions of mappings,

    N. E. Steenrod, “Products of cocycles and extensions of mappings,”Annals of Mathematics48no. 2, (1947) 290–320. Full publication date: Apr., 1947

    1947

  47. [55]

    Spin tqfts and fermionic phases of matter,

    D. Gaiotto and A. Kapustin, “Spin tqfts and fermionic phases of matter,”International Journal of Modern Physics A31no. 28n29, (2016) 1645044

    2016

  48. [56]

    Lattice models that realize𭟋 n-1 symmetry-protected topological states for evenn,

    L. Tsui and X.-G. Wen, “Lattice models that realize𭟋 n-1 symmetry-protected topological states for evenn,”Phys. Rev. B101(2020) 035101

    2020

  49. [57]

    Exact bosonization in arbitrary dimensions,

    Y.-A. Chen, “Exact bosonization in arbitrary dimensions,”Phys. Rev. Research2(2020) 033527

    2020

  50. [58]

    Fermionic SPT phases in higher dimensions and bosonization,

    A. Kapustin and R. Thorngren, “Fermionic SPT phases in higher dimensions and bosonization,” Journal of High Energy Physics2017no. 10, (2017) 80

    2017

  51. [59]

    Generalized global symmetries ofT[M] theories. part I,

    S. Gukov, P.-S. Hsin, and D. Pei, “Generalized global symmetries ofT[M] theories. part I,”Journal of High Energy Physics2021no. 4, (2021) 232

    2021

  52. [60]

    Topological Orders in (4+1)-Dimensions,

    T. Johnson-Freyd and M. Yu, “Topological Orders in (4+1)-Dimensions,”SciPost Phys.13(2022) 068

    2022

  53. [61]

    Anomalies of non-invertible symmetries in (3+1)d,

    C. C´ ordova, P.-S. Hsin, and C. Zhang, “Anomalies of non-invertible symmetries in (3+1)d,”SciPost Phys.17(2024) 131

    2024

  54. [62]

    A quantum cellular automaton for every symmetry protected topological phase,

    L. Fidkowski, J. Haah, and M. B. Hastings, “A quantum cellular automaton for every symmetry protected topological phase,”Phys. Rev. B112(2025) 035123

    2025

  55. [63]

    Anyonic membranes and pontryagin statistics,

    Y. Feng, H. Xue, Y. Li, M. Cheng, R. Kobayashi, P.-S. Hsin, and Y.-A. Chen, “Anyonic membranes and pontryagin statistics,”arXiv:2509.14314 [quant-ph]

  56. [64]

    Exact bosonization in two spatial dimensions and a new class of lattice gauge theories,

    Y.-A. Chen, A. Kapustin, and D. Radivcevi´ c, “Exact bosonization in two spatial dimensions and a new class of lattice gauge theories,”Annals Phys.393(2018) 234–253

    2018

  57. [65]

    Determination of the second homology and cohomology groups of a space by means of homotopy invariants,

    S. Eilenberg and S. MacLane, “Determination of the second homology and cohomology groups of a space by means of homotopy invariants,”Proceedings of the National Academy of Sciences of the United States of America32no. 11, (1946) 277–280

    1946

  58. [66]

    On the Groups H(Π, n), I,

    S. Eilenberg and S. MacLane, “On the Groups H(Π, n), I,”Annals of Mathematics58no. 1, (1953) 55–106. 46

    1953

  59. [67]

    On the Groups H(Π, n), II: Methods of Computation,

    S. Eilenberg and S. MacLane, “On the Groups H(Π, n), II: Methods of Computation,”Annals of Mathematics60no. 1, (1954) 49–139

    1954

  60. [68]

    In and around abelian anyon models,

    L. Wang and Z. Wang, “In and around abelian anyon models,”Journal of Physics A: Mathematical and Theoretical53no. 50, (2020) 505203

    2020

  61. [69]

    Cups and Gates I: Cohomology invariants and logical quantum operations,

    N. P. Breuckmann, M. Davydova, J. N. Eberhardt, and N. Tantivasadakarn, “Cups and Gates I: Cohomology invariants and logical quantum operations,”arXiv:2410.16250 [quant-ph]

    arXiv

  62. [70]

    Clifford Hierarchy Stabilizer Codes: Transversal Non-Clifford Gates and Magic,

    R. Kobayashi, G. Zhu, and P.-S. Hsin, “Clifford Hierarchy Stabilizer Codes: Transversal Non-Clifford Gates and Magic,”arXiv:2511.02900 [quant-ph]

    arXiv

  63. [71]

    Transversal Clifford-Hierarchy Gates via Non-Abelian Surface Codes,

    A. Warman and S. Schafer-Nameki, “Transversal Clifford-Hierarchy Gates via Non-Abelian Surface Codes,”arXiv:2512.13777 [quant-ph]

    arXiv

  64. [72]

    Surface codes: Towards practical large-scale quantum computation,

    A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: Towards practical large-scale quantum computation,”Phys. Rev. A86(2012) 032324

    2012

  65. [73]

    Codimension-2 defects and higher symmetries in (3+1)D topological phases,

    M. Barkeshli, Y.-A. Chen, S.-J. Huang, R. Kobayashi, N. Tantivasadakarn, and G. Zhu, “Codimension-2 defects and higher symmetries in (3+1)D topological phases,”SciPost Phys.14 (2023) 065

    2023

  66. [74]

    Higher-group symmetry in finite gauge theory and stabilizer codes,

    M. Barkeshli, Y.-A. Chen, P.-S. Hsin, and R. Kobayashi, “Higher-group symmetry in finite gauge theory and stabilizer codes,”SciPost Physics16no. 4, (2024)

    2024

  67. [75]

    Higher-group symmetry of (3+1)d fermionicZ 2 gauge theory: Logical ccz, cs, and t gates from higher symmetry,

    M. Barkeshli, P.-S. Hsin, and R. Kobayashi, “Higher-group symmetry of (3+1)d fermionicZ 2 gauge theory: Logical ccz, cs, and t gates from higher symmetry,”SciPost Physics16no. 5, (2024)

    2024

  68. [76]

    Classifying logical gates in quantum codes via cohomology operations and symmetry,

    P.-S. Hsin, R. Kobayashi, and G. Zhu, “Classifying logical gates in quantum codes via cohomology operations and symmetry,”arXiv:2411.15848 [quant-ph]

    arXiv

  69. [77]

    Extracting topological orders of generalized pauli stabilizer codes in two dimensions,

    Z. Liang, Y. Xu, J. T. Iosue, and Y.-A. Chen, “Extracting topological orders of generalized pauli stabilizer codes in two dimensions,”PRX Quantum5(2024) 030328

    2024

  70. [78]

    Operator algebra and algorithmic construction of boundaries and defects in (2+1)D topological Pauli stabilizer codes,

    Z. Liang, B. Yang, J. T. Iosue, and Y.-A. Chen, “Operator algebra and algorithmic construction of boundaries and defects in (2+1)D topological Pauli stabilizer codes,”arXiv:2410.11942 [quant-ph]

    arXiv

  71. [79]

    Planar quantum low-density parity-check codes with open boundaries,

    Z. Liang, J. N. Eberhardt, and Y.-A. Chen, “Planar quantum low-density parity-check codes with open boundaries,”PRX Quantum6(2025) 040330

    2025

  72. [80]

    Witt Groups and Bulk-Boundary Correspondence for Stabilizer States,

    B. Ruba and B. Yang, “Witt Groups and Bulk-Boundary Correspondence for Stabilizer States,” arXiv:2509.10418 [math-ph]

  73. [81]

    Index theory of one dimensional quantum walks and cellular automata,

    D. Gross, V. Nesme, H. Vogts, and R. F. Werner, “Index theory of one dimensional quantum walks and cellular automata,”Communications in Mathematical Physics310no. 2, (2012) 419–454

    2012

  74. [82]

    Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge,

    D. V. Else and C. Nayak, “Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge,”Phys. Rev. B90(2014) 235137

    2014

  75. [83]

    Classification of quantum cellular automata,

    M. Freedman and M. B. Hastings, “Classification of quantum cellular automata,”Communications in Mathematical Physics376no. 2, (2020) 1171–1222

    2020

  76. [84]

    Clifford quantum cellular automata: Trivial group in 2D and witt group in 3D,

    J. Haah, “Clifford quantum cellular automata: Trivial group in 2D and witt group in 3D,”Journal of Mathematical Physics62no. 9, (2021) 092202

    2021

  77. [85]

    Three-dimensional quantum cellular automata from chiral semion surface topological order and beyond,

    W. Shirley, Y.-A. Chen, A. Dua, T. D. Ellison, N. Tantivasadakarn, and D. J. Williamson, “Three-dimensional quantum cellular automata from chiral semion surface topological order and beyond,”PRX Quantum3(2022) 030326

    2022

  78. [86]

    Nontrivial quantum cellular automata in higher dimensions,

    J. Haah, L. Fidkowski, and M. B. Hastings, “Nontrivial quantum cellular automata in higher dimensions,”Communications in Mathematical Physics398no. 1, (2023) 469–540. 47

    2023

  79. [87]

    Topological phases of unitary dynamics: Classification in clifford category,

    J. Haah, “Topological phases of unitary dynamics: Classification in clifford category,” Communications in Mathematical Physics406no. 4, (2025) 76

    2025

  80. [88]

    Categorifying Clifford QCA,

    B. Yang, “Categorifying Clifford QCA,”arXiv:2504.14811 [math-ph]

  81. [89]

    Higher-form anomalies on lattices,

    Y. Feng, R. Kobayashi, Y.-A. Chen, and S. Ryu, “Higher-form anomalies on lattices,” arXiv:2509.12304 [cond-mat.str-el]

  82. [90]

    Onsiteability of higher-form symmetries,

    Y. Feng, Y.-A. Chen, P.-S. Hsin, and R. Kobayashi, “Onsiteability of higher-form symmetries,” arXiv:2510.23701 [cond-mat.str-el]

    arXiv

  83. [91]

    Higher symmetries and anomalies in quantum lattice systems,

    A. Kapustin and S. Xu, “Higher symmetries and anomalies in quantum lattice systems,” arXiv:2505.04719 [math-ph]

    arXiv

  84. [92]

    Higher symmetries, anomalies, and crossed squares in lattice gauge theory,

    A. Kapustin and L. Spodyneiko, “Higher symmetries, anomalies, and crossed squares in lattice gauge theory,”arXiv:2507.16966 [hep-th]

    arXiv

  85. [93]

    Anomalies on the Lattice, Homotopy of Quantum Cellular Automata, and a Spectrum of Invertible States,

    A. M. Czajka, R. Geiko, and R. Thorngren, “Anomalies on the Lattice, Homotopy of Quantum Cellular Automata, and a Spectrum of Invertible States,”arXiv:2512.02105 [cond-mat.str-el]. 48

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Reviewed August 3, 2026 · model on record in the stance chip above.