## Description

Also known as the \(\mathbb{Z}_q\) surface code. Extension of the surface code to prime-dimensional [1][2] and more general modular qudits [3]. Stabilizer generators are few-body \(X\)-type and \(Z\)-type Pauli strings associated to the stars and plaquettes, respectively, of a tessellation of a two-dimensional surface. Since qudits have more than one \(X\) and \(Z\)-type operator, various sets of stabilizer generators can be defined. Ground-state degeneracy and the associated phase depends on the qudit dimension and the stabilizer generators.

## Decoding

Renormalization-group decoder [4].

## Notes

The simplest Decodoku game is based on the qudit surface code with \( q=10\). See related Qiskit tutorial.

## Parent

- Modular-qudit CSS code — Plaquette and star operators are stabilizer generators.

## Cousins

- Abelian topological code — Modular-qudit surface code Hamiltonians can admit topological phases associated with \(\mathbb{Z}_q\) [2].
- Kitaev surface code — The qudit surface code with \(q=2\) is the surface code.
- String-net code — String-net model reduces to the qudit surface code when the category is the group \(\mathbb{Z}_q\).
- Quantum-double code — A quantum-double model with \(G=\mathbb{Z}_q\) is the qudit surface code.
- 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 [5].
- \(\mathbb{Z}_4^{(1)}\) subsystem code — The \(\mathbb{Z}_4^{(1)}\) subsystem code can be obtained from the \(\mathbb{Z}_4\) surface code by gauging out the anyon \(e m^3\) [6; Fig. 15].
- Semion subsystem code — The anyon fusion rules for the double-semion code and the \(\mathbb{Z}_4\) surface code are the same, but exchange statistics are different. The double-semion code can be obtained from the \(\mathbb{Z}_4\) surface code by condensing the anyon \(e^2 m^2\) [6] or by gauging out the 1-form symmetry associated with said anyon [6; Footnote 18].
- Double-semion stabilizer code — The anyon fusion rules for the double-semion code and the \(\mathbb{Z}_4\) surface code are the same, but exchange statistics are different. The double-semion code can be obtained from the \(\mathbb{Z}_4\) surface code by condensing the anyon \(e^2 m^2\) [6] or by gauging out the 1-form symmetry associated with said anyon [6; Footnote 18].

## References

- [1]
- A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003) arXiv:quant-ph/9707021 DOI
- [2]
- 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
- [3]
- H. Watanabe, M. Cheng, and Y. Fuji, “Ground state degeneracy on torus in a family of \(\mathbb{Z}_N\) toric code”, (2022) arXiv:2211.00299
- [4]
- H. Anwar et al., “Fast decoders for qudit topological codes”, New Journal of Physics 16, 063038 (2014) arXiv:1311.4895 DOI
- [5]
- J. Haah, “Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices”, Journal of Mathematical Physics 62, 012201 (2021) arXiv:1812.11193 DOI
- [6]
- T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, (2022) arXiv:2211.03798

## Page edit log

- Victor V. Albert (2022-11-02) — most recent
- Victor V. Albert (2022-01-05)
- Ian Teixeira (2021-12-19)

## Cite as:

“Modular-qudit surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudit_surface