# 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.

There exists an invariant that can be computed to uniquely characterize the anyons of a state in an Abelian quantum-double topological phase [2].

## Protection

Error-correcting properties established in Ref. [3] using operator algebra theory. Correcting the maximum number of correctable errors is \(NP\)-complete [4].

## Encoding

Any geometrically local unitary circuit connecting two quantum double models whose groups are not isomorphic must have depth at linear linear in \(n\) [2].

## Decoding

Efficient decoder correcting below the code distance [4].

## 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 [5]. Upon gauging some symmetries [6–8,8], Type-I and II \(\mathbb{Z}_2^3\) TQDs realize the same topological order as certain Abelian quantum double models [9–11].
- Quantum-double code — The anyon theory corresponding to (Abelian) quantum double codes is defined by an (Abelian) group.

## Children

- 2D color code — When treated as ground states of the code Hamiltonian, states of the color code on a torus geometry realize \(\mathbb{Z}_2\times\mathbb{Z}_2\) topological order [12], equivalent to the phase realized by two copies of the surface code via a local constant-depth Clifford circuit [13]. This process can be viewed as an ungauging [6–8,8] of certain symmetries.
- Matching code — Matching codes were inspired by the \(\mathbb{Z}_2\) topological order phase of the Kitaev honeycomb model [14].
- 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 [15]. Local Clifford deformations preserve this topological order.
- XY surface code
- XZZX surface code
- Modular-qudit surface code — Modular-qudit surface code Hamiltonians admit topological phases associated with \(\mathbb{Z}_q\) topological order [16].

## Cousins

- 2D lattice stabilizer code — Translation-invariant 2D prime-qudit lattice stabilizer codes are equivalent to several copies of the prime-qudit surface code and a trivial code via a local constant-depth Clifford circuit [17].
- XZZX surface code — The XZZX surface code is an example of \(\mathbb{Z}_2\) topological order as manifest in the Wen plaquette model [18].
- 3D subsystem surface code — The 3D subsystem surface code Hamiltonian phase diagram exhibits \(\mathbb{Z}_2\) topological order [19].
- \(\mathbb{Z}_3\times\mathbb{Z}_9\)-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 \(m_2^{-1}\) [5; Sec. 7.5].
- Galois-qudit color code — A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [20,22][21; Sec. 5.3]. Galois-qudit color codes yield Abelian quantum-double codes with Abelian-group topological order via this decomposition.
- Galois-qudit surface code — A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [20,22][21; Sec. 5.3]. Galois-qudit surface codes yield Abelian quantum-double codes with \(GF(p^m)\cong \mathbb{Z}_p^m\) topological order 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]
- J. Haah, “An Invariant of Topologically Ordered States Under Local Unitary Transformations”, Communications in Mathematical Physics 342, 771 (2016) arXiv:1407.2926 DOI
- [3]
- 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
- [4]
- S. X. Cui, C. Galindo, and D. Romero, “Abelian Group Quantum Error Correction in Kitaev’s Model”, (2024) arXiv:2404.08552
- [5]
- T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
- [6]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
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- L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter”, Journal of High Energy Physics 2017, (2017) arXiv:1605.01640 DOI
- [8]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
- [9]
- M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
- [10]
- B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI
- [11]
- 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
- [12]
- 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
- [13]
- A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
- [14]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
- [15]
- F. J. Wegner, “Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters”, Journal of Mathematical Physics 12, 2259 (1971) DOI
- [16]
- 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
- [17]
- 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
- [18]
- X.-G. Wen, “Quantum Orders in an Exact Soluble Model”, Physical Review Letters 90, (2003) arXiv:quant-ph/0205004 DOI
- [19]
- Y. Li et al., “Phase diagram of the three-dimensional subsystem toric code”, Physical Review Research 6, (2024) arXiv:2305.06389 DOI
- [20]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [21]
- A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
- [22]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL

## 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