# Abelian quantum double stabilizer code[1]

## Description

Modular-qudit stabilizer code whose codewords realize 2D modular gapped Abelian topological order with trivial cocycle. The corresponding anyon theory is defined by an Abelian group. All such codes can be realized by a stack of modular-qudit surface codes because all Abelian groups are Kronecker products of cyclic groups.

## Protection

Error-correcting properties established in Ref. [2] using operator algebra theory.

## Parents

- Abelian TQD stabilizer code — The anyon theory corresponding to Abelian quantum double codes is defined by an Abelian group and trivial cocycle. All Abelian TQD codes can be realized as modular-qudit stabilizer codes by starting with an Abelian quantum double model along with a family of Abelian TQDs that generalize the double semion anyon theory and condensing certain bosonic anyons [3]. Upon gauging, Type-I and II \(\mathbb{Z}_2^3\) TQDs realize the same topological order as certain Abelian quantum double models [4–6].
- Quantum-double code — The anyon theory corresponding to (Abelian) quantum double codes is defined by an (Abelian) group.

## Children

- Two-dimensional color code — When treated as ground states of the code Hamiltonian, states of the color code on a torus geometry on realize \(\mathbb{Z}_2\times\mathbb{Z}_2\) topological order [7], equivalent to the phase realized by two copies of the surface code [8].
- Matching code — Matching codes were inspired by the \(\mathbb{Z}_2\) topological order phase of the Kitaev honeycomb model [9].
- Clifford-deformed surface code (CDSC) — When treated as ground states of the code Hamiltonian, surface codewords realize \(\mathbb{Z}_2\) topological order, a topological phase of matter that also exists in \(\mathbb{Z}_2\) lattice gauge theory [10]. Local Clifford deformations preserve this topological order.
- Modular-qudit surface code — Modular-qudit surface code Hamiltonians admit topological phases associated with \(\mathbb{Z}_q\) [11].

## Cousins

- Translationally invariant stabilizer code — Translation-invariant 2D prime-qudit topological stabilizer codes are equivalent to several copies of the prime-qudit surface code via a local constant-depth Clifford circuit [12].
- XZZX surface code — Example of \(\mathbb{Z}_2\) topological order as manifest in the Wen plaquette model [13].
- 3D subsystem surface code — The 3D subsystem surface code Hamiltonian phase diagram exhibits \(\mathbb{Z}_2\) topological order [14].
- \(\mathbb{Z}_3\times\mathbb{Z}_9\)-fusion subsystem code — The \(\mathbb{Z}_3\times\mathbb{Z}_9\)-fusion subsystem code can be obtained from a stack of \(q=3\) and \(q=9\) square-lattice qudit surface codes by gauging out the anyons \(m_1^{-1}e_2^3\) and \(m_2^{-1}\) [3; Sec. 7.5].
- Galois-qudit topological code — A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [15]; see Sec. 5.3 of Ref. [16]. Galois-qudit topological surface and color codes yield Abelian quantum-double codes via this decomposition.

## References

- [1]
- A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003) arXiv:quant-ph/9707021 DOI
- [2]
- M. Cha, P. Naaijkens, and B. Nachtergaele, “On the Stability of Charges in Infinite Quantum Spin Systems”, Communications in Mathematical Physics 373, 219 (2019) arXiv:1804.03203 DOI
- [3]
- T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
- [4]
- M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
- [5]
- B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI
- [6]
- L. Lootens et al., “Mapping between Morita-equivalent string-net states with a constant depth quantum circuit”, Physical Review B 105, (2022) arXiv:2112.12757 DOI
- [7]
- 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) arXiv:0906.4127 DOI
- [8]
- A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
- [9]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
- [10]
- F. J. Wegner, “Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters”, Journal of Mathematical Physics 12, 2259 (1971) DOI
- [11]
- 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) arXiv:quant-ph/0609070 DOI
- [12]
- J. Haah, “Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices”, Journal of Mathematical Physics 62, (2021) arXiv:1812.11193 DOI
- [13]
- X.-G. Wen, “Quantum Orders in an Exact Soluble Model”, Physical Review Letters 90, (2003) arXiv:quant-ph/0205004 DOI
- [14]
- Y. Li et al., “Phase diagram of the three-dimensional subsystem toric code”, (2023) arXiv:2305.06389
- [15]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [16]
- A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074

## Page edit log

- Victor V. Albert (2023-04-06) — most recent

## Cite as:

“Abelian quantum double stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/quantum_double_abelian