# Abelian topological code

## Description

Code whose codewords realize topological order associated with an Abelian anyon theory, i.e., an anyon theory which generates an Abelian group under fusion. Any theory is defined by an Abelian group and, for each anyon type \(a\), the phases \(\theta(a)\) obtained by exchanging two anyons of said type CITE In and around Abelian anyon models.

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].

## Gates

Clifford gates can be implemented by braiding defects; for qubit-based stabilizer codes realizing Abelian topological phases, see Refs. [2][3]. Most of such designs focus on the surface code [4][5][6][7][8][9].

## Fault Tolerance

Fault-tolerant logical operations can be interpreted as anyon condensation events [10].Modular decoding applicable to all fault-tolerant protocols based on topological qubit stabilizer codes [11].

## Code Capacity Threshold

Noise thresholds can be formulated as anyon condensation transitions in a topological field theory [12], generalizing the mapping of the effect of noise on a code state to a statistical mechanical model [13][14][15][16].

## Parents

- Subsystem qubit stabilizer code — All premodular Abelian topological orders can be realized as modular-qudit subsystem stabilizer codes whose stabilizer generators may not be geometrically local [17].
- Topological code

## Children

- 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 [18], equivalent to the phase realized by two copies of the surface code [19].
- Matching code — Matching codes were inspired by the \(\mathbb{Z}_2\) topological order phase of the Kitaev honeycomb model [20].
- 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 [21]. Local Clifford deformations preserve this topological order.
- \(\mathbb{Z}_4^{(1)}\) subsystem code — When treated as ground states of the code Hamiltonian, the code states realize \(\mathbb{Z}_4^{(1)}\) topological order [22].
- Semion subsystem code — When treated as ground states of the code Hamiltonian, the code states realize chiral-semion topological order [17].
- Double-semion stabilizer 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 [23].
- Modular-qudit surface code — Modular-qudit surface code Hamiltonians can admit topological phases associated with \(\mathbb{Z}_q\) [24].
- Galois-qudit topological code

## Cousins

- Modular-qudit stabilizer code — All Abelian topological orders admitting a gapped boundary can be realized as modular-qudit stabilizer codes [25].
- Modular-qudit surface code — All premodular Abelian topological orders can be realized with a stack of modular-qudit surface codes and a family of twisted quantum doubles that generalize the double semion anyon theory [17].
- Double-semion stabilizer code — All premodular Abelian topological orders can be realized with a stack of modular-qudit surface codes and a family of twisted quantum doubles that generalize the double semion anyon theory [17].
- 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 [26].
- XZZX surface code — Example of \(\mathbb{Z}_2\) topological order in the Wen plaquette model [27].

## References

- [1]
- J. Haah, “An Invariant of Topologically Ordered States Under Local Unitary Transformations”, Communications in Mathematical Physics 342, 771 (2016) arXiv:1407.2926 DOI
- [2]
- 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
- [3]
- Y. D. Lensky et al., “Graph gauge theory of mobile non-Abelian anyons in a qubit stabilizer code”, (2022) arXiv:2210.09282
- [4]
- H. Bombin, “Topological Order with a Twist: Ising Anyons from an Abelian Model”, Physical Review Letters 105, (2010) arXiv:1004.1838 DOI
- [5]
- A. Kitaev and L. Kong, “Models for Gapped Boundaries and Domain Walls”, Communications in Mathematical Physics 313, 351 (2012) arXiv:1104.5047 DOI
- [6]
- A. G. Fowler et al., “Surface codes: Towards practical large-scale quantum computation”, Physical Review A 86, (2012) arXiv:1208.0928 DOI
- [7]
- 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
- [8]
- 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
- [9]
- 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
- [10]
- M. S. Kesselring et al., “Anyon condensation and the color code”, (2022) arXiv:2212.00042
- [11]
- H. Bombín et al., “Modular decoding: parallelizable real-time decoding for quantum computers”, (2023) arXiv:2303.04846
- [12]
- Y. Bao et al., “Mixed-state topological order and the errorfield double formulation of decoherence-induced transitions”, (2023) arXiv:2301.05687
- [13]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [14]
- A. A. Kovalev and L. P. Pryadko, “Fault tolerance of quantum low-density parity check codes with sublinear distance scaling”, Physical Review A 87, (2013) arXiv:1208.2317 DOI
- [15]
- A. A. Kovalev and L. P. Pryadko, “Spin glass reflection of the decoding transition for quantum error correcting codes”, (2014) arXiv:1311.7688
- [16]
- C. T. Chubb and S. T. Flammia, “Statistical mechanical models for quantum codes with correlated noise”, Annales de l’Institut Henri Poincaré D 8, 269 (2021) arXiv:1809.10704 DOI
- [17]
- T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, (2022) arXiv:2211.03798
- [18]
- 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
- [19]
- A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
- [20]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
- [21]
- F. J. Wegner, “Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters”, Journal of Mathematical Physics 12, 2259 (1971) DOI
- [22]
- P. H. Bonderson, Non-Abelian Anyons and Interferometry, California Institute of Technology, 2007 DOI
- [23]
- R. Dijkgraaf and E. Witten, “Topological gauge theories and group cohomology”, Communications in Mathematical Physics 129, 393 (1990) DOI
- [24]
- 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
- [25]
- T. D. Ellison et al., “Pauli Stabilizer Models of Twisted Quantum Doubles”, PRX Quantum 3, (2022) arXiv:2112.11394 DOI
- [26]
- X. Ni, O. Buerschaper, and M. Van den Nest, “A non-commuting stabilizer formalism”, Journal of Mathematical Physics 56, 052201 (2015) arXiv:1404.5327 DOI
- [27]
- X.-G. Wen, “Quantum Orders in an Exact Soluble Model”, Physical Review Letters 90, (2003) arXiv:quant-ph/0205004 DOI

## 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