# Abelian topological code

## Description

Code whose codewords realize topological order associated with an abelian group. Stub.

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 system’s diameter [1].

## Parents

- Topological code
- Quantum low-density parity-check (QLDPC) code — All abelian topological orders can be realized as geometrically local modular-qudit stabilizer codes [2], and topological-code Hamiltonians are geometrically local for appropriate tesselations.

## Children

- Clifford-deformed surface code (CDSC) — Local deformations of the surface code preserve its \(\mathbb{Z}_2\) topological order.
- Color code — When treated as ground states of the code Hamiltonian, 2D color code states on realize \(\mathbb{Z}_2\times\mathbb{Z}_2\) topological order [3], equivalent to the phase realized by two copies of the surface code [4].
- Double-semion code — When treated as ground states of the code Hamiltonian, the code states realize double-semion topological order, a topological phase of matter that also exists in twisted \(\mathbb{Z}_2\) gauge theory [5].
- Kitaev surface code — When treated as ground states of the code Hamiltonian, the code states realize \(\mathbb{Z}_2\) topological order, a topological phase of matter that also exists in \(\mathbb{Z}_2\) lattice gauge theory [6].
- Matching code — Matching codes were inspired by the Kitaev honeycomb model [7], which realizes \(\mathbb{Z}_2\) topological order.
- Modular-qudit surface code — Qudit surface code Hamiltonians admit topological phases associated with \(\mathbb{Z}_q\) [8].

## Cousins

- Modular-qudit stabilizer code — All abelian topological orders can be realized as modular-qudit stabilizer codes [2].
- Translationally-invariant stabilizer code — Translationally-invariant stabilizer codes can realize abelian topological orders. Conversely, abelian topological codes need not be translationally invariant, and can realize multiple topological phases on one lattice.
- XS stabilizer code — Twisted quantum double models for the groups \(\mathbb{Z}_2^k\) can be realized as XS stabilizer codes.
- XZZX surface code — Example of \(\mathbb{Z}_2\) topological order in the Wen plaquette model [9].

## Zoo code information

## References

- [1]
- J. Haah, “An Invariant of Topologically Ordered States Under Local Unitary Transformations”, Communications in Mathematical Physics 342, 771 (2016). DOI; 1407.2926
- [2]
- Tyler D. Ellison et al., “Pauli stabilizer models of twisted quantum doubles”. 2112.11394
- [3]
- M. Kargarian, H. Bombin, and M. A. Martin-Delgado, “Topological color codes and two-body quantum lattice Hamiltonians”, New Journal of Physics 12, 025018 (2010). DOI; 0906.4127
- [4]
- A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015). DOI; 1503.02065
- [5]
- R. Dijkgraaf and E. Witten, “Topological gauge theories and group cohomology”, Communications in Mathematical Physics 129, 393 (1990). DOI
- [6]
- F. J. Wegner, “Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters”, Journal of Mathematical Physics 12, 2259 (1971). DOI
- [7]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006). DOI; cond-mat/0506438
- [8]
- S. S. Bullock and G. K. Brennen, “Qudit surface codes and gauge theory with finite cyclic groups”, Journal of Physics A: Mathematical and Theoretical 40, 3481 (2007). DOI; quant-ph/0609070
- [9]
- X.-G. Wen, “Quantum Orders in an Exact Soluble Model”, Physical Review Letters 90, (2003). DOI; quant-ph/0205004

## Cite as:

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