Abelian topological code 


Code whose codewords realize topological order associated with an Abelian anyon theory. In 2D, this is equivalent to a unitary braided fusion category which is also an Abelian group under fusion [1]. Unless otherwise noted, the phases discussed are bosonic.

2D Abelian topological codes

A theory is defined by an Abelian group \(A\) of anyon types whose multiplication relations define the fusion rules, and a set of exchange statistics \(\theta(a)\in U(1)\) obtained by exchanging two anyons of type \(a\in A\). The exchange statistics in turn define braiding relations, \begin{align} B(a,b) = \frac{\theta(ab)}{\theta(a)\theta(b)}~, \tag*{(1)}\end{align} between all anyon pairs \(a,b\).

All 2D Abelian bosonic topological orders can be understood within the subsystem stabilizer formalism [2]. As such, many of the operations one can perform on such codes have both a stabilizer and a topological-phase interpretation. Stabilizer generators of 2D topological codes acting on 1D loops of qubits can be interpreted as one-form symmetries of the underlying phase realized by the code. Identification of an anyon \(a\) with the vacuum is equivalent to adding string excitation operators corresponding to \(a\) to the stabilizer group and taking the center to get another stabilizer group. Code states of this new stabilizer code correspond to a condensed phase of the parent topological phase. The remaining unidentified parent-phase anyons behave differently with respect to the new condensed-phase state. Some become confined while the remaining ones pick up new braiding relations.

Gapped anyon theories admit a subgroup of bosons with trivial mutual statistics whose order squares to that of \(G\); see Ref. [3]. In terms of their category theoretic structure, gapped anyon theories admit a Lagrangian subgroup [4,5].

3D abelian bosonic topological codes

There are three types of \(\mathbb{Z}_2\) topological orders in 3D: one with bosonic charge and loop excitations (BcBl) and two with fermionic charge excitations and bosonic (FcBl) and fermionic (FcFl) loop excitations, respectively [6,7]. There exists an invariant that distinguishes these [7]. A similar pattern follows in higher dimensions.


Any local quantum circuit connecting ground states of topological orders with non-isomorphic Abelian groups must have depth that is at least linear in the diameter of the system [8].


Clifford gates can be implemented by braiding defects; for qubit-based stabilizer codes realizing Abelian topological phases, see Refs. [9,10]. Most of such designs focus on the surface code [1116].

Fault Tolerance

Fault-tolerant logical operations can be interpreted as anyon condensation events [17].Modular decoding, designed to overcome the backlog problem, is applicable to fault-tolerant protocols based on topological qubit stabilizer codes [18].



  • Majorana box qubit — When treated as ground states of the code Hamiltonian, surface codewords realize, codewords of a single Kitaev chain realize \(\mathbb{Z}_2\) fermionic topological order. The MZMs used to define the tetron code act as Ising anyons, which are non-Abelian.
  • Layer code — The Layer code realizes 2D layers of \(\mathbb{Z}_2\) gauge theory coupled along defects.
  • 3D color code
  • 3D fermionic surface code — The 3D Kitaev surface code realizes 3D \(\mathbb{Z}_2\) gauge theory with fermionic charge and bosonic loop excitations (FcBl), i.e., with an emergent fermion.
  • 3D surface code — The 3D Kitaev surface code realizes 3D \(\mathbb{Z}_2\) gauge theory with bosonic charge and loop excitations (BcBl). The welded surface code does not satisfy homogeneous topological order [19].
  • \((1,3)\) 4D toric code — The \((1,3)\) 4D toric code realizes 4D \(\mathbb{Z}_2\) gauge theory with 1D \(Z\)-type and 3D \(X\)-type logical operators.
  • Loop toric code — The 4D loop toric code realizes 4D \(\mathbb{Z}_2\) gauge theory with only loop excitations [20].
  • Three-fermion (3F) Walker-Wang model code — When treated as ground states of the code Hamiltonian, 3F model code states realize 3F topological order, which is chiral and modular.
  • Three-fermion (3F) subsystem code — The 3F code is a 2D subsystem code characterized by 3F topological order [2], which is chiral and modular.
  • Abelian TQD stabilizer code — Abelian TQDs realize all modular gapped Abelian topological orders [3]. Conversely, every Abelian anyon theory is a subtheory of some TQD [2; Sec. 6.2]. Any Abelian anyon theory \(A\) can be realized at one of the surfaces of a 3D Walker-Wang model whose underlying theory is an Abelian TQD containing \(A\) as a subtheory [21,22][2; Appx. H].
  • \(\mathbb{Z}_q^{(1)}\) subsystem code — The \(\mathbb{Z}_q^{(1)}\) subsystem code is characterized by the \(\mathbb{Z}_q^{(1)}\) anyon theory [23]. The anyon theory has a single generator \(a \in \mathbb Z_N\) with \(\theta(a) =e^{\frac{2\pi i}{N}a^2}\). It is modular for odd prime \(q\) and non-modular otherwise.
  • Chiral semion subsystem code — The semion code is a subsystem code characterized by the chiral semion topological phase.
  • \(\mathbb{Z}_3\times\mathbb{Z}_9\)-fusion subsystem code — The \(\mathbb{Z}_q^{(1)}\) subsystem code is characterized by a non-modular anyon theory with \(\mathbb{Z}_3\times\mathbb{Z}_9\) fusion rules.
  • Galois-qudit topological code


  • Hamiltonian-based code — Subsystem stabilizer code Hamiltonians described by an Abelian anyon theory do not always realize the corresponding anyonic topological order in their ground-state subspace and may exhibit a rich phase diagram. For example, the Kitaev honeycomb Hamiltonian admits the anyon theories of the 16-fold way, i.e., all minimal modular extensions of the \(\mathbb{Z}_2^{(1)}\) Abelian non-chiral non-modular anyon theory [24][2; Footnote 25].
  • Walker-Wang model code — Any Abelian anyon theory \(A\) can be realized at one of the surfaces of a 3D Walker-Wang model whose underlying theory is an Abelian TQD containing \(A\) as a subtheory [21,22][2; Appx. H].
  • 3D lattice stabilizer code — Qubit 3D stabilizer codes are conjectured to admit either fracton phases or abelian topological phases that are equivalent to multiple copies of the 3D surface code and/or the 3D fermionic surface code [25].
  • Lattice subsystem code — All 2D Abelian bosonic topological orders can be realized as modular-qudit lattice subsystem codes by starting with an Abelian quantum double model (slightly different from that of Ref. [3]) along with a family of Abelian TQDs that generalize the double semion anyon theory and gauging out certain bosonic anyons [2]. The stabilizer generators of the new subsystem code may no longer be geometrically local. Non-Abelian topological orders are purported not to be realizable with Pauli stabilizer codes [26].
  • Gauss' law code — Gauge-group elements of a \(D\)-dimensional fermionic \(\mathbb{Z}_2\) gauge theory form a single-error-correcting linear binary code [27; Thm. 1].
  • Classical topological code — Some topological orders have classical analogues that can be used for error correction.
  • Analog surface code — The analog surface code realizes a straightforward extension of the modular-qudit surface code to infinite local dimension, \(q\to\infty\). There are two types of anyons, \(e\) and \(m\), with each type being valued in \(U(1)\) as opposed to \(\mathbb{Z}_q\) for the qudit surface code.
  • Dynamical automorphism (DA) code — Useful measurement sequences of DA codes can be extracted from topological quantum field theory [28].
  • Quantum repetition code — The 1D quantum repetition code is an ingredient in product constructions that yield several topological phases [29; Fig. 8].
  • Twist-defect color code — Twist-defect color codes realize \(\mathbb{Z}_2 \times \mathbb{Z}_2\) topological order with twist defects.
  • Chen-Hsin invertible-order code — Instances of the code in 4D realize the 3D \(\mathbb{Z}_2\) gauge theory with fermionic charge and either bosonic (FcBl) or fermionic (FcFl) loop excitations at their boundaries [6,7]; see Ref. [30] for a different lattice-model formulation of the FcBl boundary code.
  • Twist-defect surface code — Twist-defect surface codes realize \(\mathbb{Z}_2\) topological order with twist defects.


L. Wang and Z. Wang, “In and around abelian anyon models \({}^{\text{*}}\)”, Journal of Physics A: Mathematical and Theoretical 53, 505203 (2020) arXiv:2004.12048 DOI
T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
T. D. Ellison et al., “Pauli Stabilizer Models of Twisted Quantum Doubles”, PRX Quantum 3, (2022) arXiv:2112.11394 DOI
A. Kapustin and N. Saulina, “Topological boundary conditions in abelian Chern–Simons theory”, Nuclear Physics B 845, 393 (2011) arXiv:1008.0654 DOI
J. Kaidi et al., “Higher central charges and topological boundaries in 2+1-dimensional TQFTs”, SciPost Physics 13, (2022) arXiv:2107.13091 DOI
T. Johnson-Freyd, “(3+1)D topological orders with only a \(\mathbb{Z}_2\)-charged particle”, (2020) arXiv:2011.11165
L. Fidkowski, J. Haah, and M. B. Hastings, “Gravitational anomaly of (3+1) -dimensional Z2 toric code with fermionic charges and fermionic loop self-statistics”, Physical Review B 106, (2022) arXiv:2110.14654 DOI
J. Haah, “An Invariant of Topologically Ordered States Under Local Unitary Transformations”, Communications in Mathematical Physics 342, 771 (2016) arXiv:1407.2926 DOI
M. Barkeshli, C.-M. Jian, and X.-L. Qi, “Theory of defects in Abelian topological states”, Physical Review B 88, (2013) arXiv:1305.7203 DOI
Y. D. Lensky et al., “Graph gauge theory of mobile non-Abelian anyons in a qubit stabilizer code”, Annals of Physics 452, 169286 (2023) arXiv:2210.09282 DOI
H. Bombin, “Topological Order with a Twist: Ising Anyons from an Abelian Model”, Physical Review Letters 105, (2010) arXiv:1004.1838 DOI
A. Kitaev and L. Kong, “Models for Gapped Boundaries and Domain Walls”, Communications in Mathematical Physics 313, 351 (2012) arXiv:1104.5047 DOI
A. G. Fowler et al., “Surface codes: Towards practical large-scale quantum computation”, Physical Review A 86, (2012) arXiv:1208.0928 DOI
H. Zheng, A. Dua, and L. Jiang, “Demonstrating non-Abelian statistics of Majorana fermions using twist defects”, Physical Review B 92, (2015) arXiv:1508.04166 DOI
B. J. Brown et al., “Poking Holes and Cutting Corners to Achieve Clifford Gates with the Surface Code”, Physical Review X 7, (2017) arXiv:1609.04673 DOI
A. Benhemou, J. K. Pachos, and D. E. Browne, “Non-Abelian statistics with mixed-boundary punctures on the toric code”, Physical Review A 105, (2022) arXiv:2103.08381 DOI
M. S. Kesselring et al., “Anyon Condensation and the Color Code”, PRX Quantum 5, (2024) arXiv:2212.00042 DOI
H. Bombín et al., “Modular decoding: parallelizable real-time decoding for quantum computers”, (2023) arXiv:2303.04846
J. Haah, “A degeneracy bound for homogeneous topological order”, SciPost Physics 10, (2021) arXiv:2009.13551 DOI
X. Chen et al., “Loops in 4+1d topological phases”, SciPost Physics 15, (2023) arXiv:2112.02137 DOI
J. Haah, “Clifford quantum cellular automata: Trivial group in 2D and Witt group in 3D”, Journal of Mathematical Physics 62, (2021) arXiv:1907.02075 DOI
W. Shirley et al., “Three-Dimensional Quantum Cellular Automata from Chiral Semion Surface Topological Order and beyond”, PRX Quantum 3, (2022) arXiv:2202.05442 DOI
P. H. Bonderson, Non-Abelian Anyons and Interferometry, California Institute of Technology, 2007 DOI
A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
A. Dua et al., “Sorting topological stabilizer models in three dimensions”, Physical Review B 100, (2019) arXiv:1908.08049 DOI
A. C. Potter and R. Vasseur, “Symmetry constraints on many-body localization”, Physical Review B 94, (2016) arXiv:1605.03601 DOI
L. Spagnoli, A. Roggero, and N. Wiebe, “Fault-tolerant simulation of Lattice Gauge Theories with gauge covariant codes”, (2024) arXiv:2405.19293
M. Davydova et al., “Quantum computation from dynamic automorphism codes”, (2023) arXiv:2307.10353
T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes II. Product constructions”, (2024) arXiv:2402.16831
L. Fidkowski, J. Haah, and M. B. Hastings, “Exactly solvable model for a 4+1D beyond-cohomology symmetry-protected topological phase”, Physical Review B 101, (2020) arXiv:1912.05565 DOI
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“Abelian topological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/topological_abelian
@incollection{eczoo_topological_abelian, title={Abelian topological code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/topological_abelian} }
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“Abelian topological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/topological_abelian

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