## Description

Modular-qudit subsystem stabilizer code, based on the Kitaev honeycomb model [3] and its generalization [1], that is characterized by \(\mathbb{Z}_q^{(1)}\) topological order [4], which is modular for odd prime \(q\) and non-modular otherwise. Encodes a single \(q\)-dimensional qudit when put on a torus for odd \(q\), and a \(q/2\)-dimensional qudit for even \(q\). This code can be constructed using geometrically local gauge generators, but does not admit geometrically local stabilizer generators. For \(q=2\), the code reduces to the subsystem code underlying the Kitaev honeycomb model code as well as the honeycomb Floquet code.

## Parent

- Abelian topological code — The \(\mathbb{Z}_q^{(1)}\) subsystem code is characterized by \(\mathbb{Z}_q^{(1)}\) topological order [4], which is modular for odd prime \(q\) and non-modular otherwise.

## Children

- Honeycomb Floquet code — The dynamically generated logical qubit of the honeycomb Floquet code is generated by appropriately scheduling measurements of the gauge generators of the \(\mathbb{Z}_{q=2}^{(1)}\) subsystem stabilizer code corresponding to the Kitaev honeycomb model.
- Kitaev honeycomb code — The Kitaev honeycomb model can be formulated as a qubit subsystem stabilizer code based on the \(\mathbb{Z}_{q=2}^{(1)}\) abelian anyon theory, which is non-chiral and non-modular [2; Sec. 7.3].

## Cousins

- Modular-qudit surface code — The \(\mathbb{Z}_q^{(1)}\) subsystem code can be obtained from the \(\mathbb{Z}_q\) square-lattice surface code by gauging out the anyon \(e^{-1} m\) and applying single-qubit Clifford gates [2; Sec. 7.3]. During this process, the square lattice is effectively expanded to a hexagonal lattice [2; Fig. 12].
- Double-semion stabilizer code — The anyonic exchange statistics of \(\mathbb{Z}_4^{(1)}\) subsystem code resemble those of the double semion code, but its fusion rules realize the \(\mathbb{Z}_4\) group.
- Chiral semion subsystem code — The semion code can be obtained from the \(\mathbb{Z}_4^{(1)}\) subsystem code by condensing the anyon \(s^2\) [2; Fig. 15].

## References

- [1]
- M. Barkeshli et al., “Generalized Kitaev Models and Extrinsic Non-Abelian Twist Defects”, Physical Review Letters 114, (2015) arXiv:1405.1780 DOI
- [2]
- T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, (2022) arXiv:2211.03798
- [3]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
- [4]
- P. H. Bonderson, Non-Abelian Anyons and Interferometry, California Institute of Technology, 2007 DOI

## Page edit log

- Nathanan Tantivasadakarn (2023-04-08) — most recent
- Victor V. Albert (2023-04-08)
- Victor V. Albert (2023-02-07)

## Cite as:

“\(\mathbb{Z}_q^{(1)}\) subsystem code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/qudit_znone