## Description

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.

## Encoding

## Gates

## Fault Tolerance

## Parent

## Children

- 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

## Cousins

- 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.

## References

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## Page edit log

- Victor V. Albert (2022-02-08) — most recent

## Cite as:

“Abelian topological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/topological_abelian