## Description

Modular-qudit subsystem code, based on the Kitaev honeycomb model [3] and its generalization [1], that is characterized by the \(\mathbb{Z}_q^{(1)}\) anyon theory [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.

## Parents

- Subsystem modular-qudit stabilizer code
- Lattice subsystem code
- Abelian topological code — The \(\mathbb{Z}_q^{(1)}\) subsystem code is characterized by the \(\mathbb{Z}_q^{(1)}\) anyon theory [4]. The anyon theory has a single generator \(a \in \mathbb Z_N\) with \(\theta(a) =e^{\frac{2\pi i}{N}a^2}\). It is modular for odd prime \(q\) and non-modular otherwise.

## Child

- Kitaev honeycomb code — The Kitaev honeycomb model can be formulated as a qubit subsystem stabilizer code [5; Sec. 3.2] 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.
- 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. However, since this subsystem code has zero logical qubits, the instantaneous stabilizer codes of the honeycomb code cannot be interpreted as gauge-fixed versions of this subsystem code.
- 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”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
- [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
- [5]
- M. Suchara, S. Bravyi, and B. Terhal, “Constructions and noise threshold of topological subsystem codes”, Journal of Physics A: Mathematical and Theoretical 44, 155301 (2011) arXiv:1012.0425 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