Also known as \(\mathbb{Z}_q\) surface code.

## Description

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

## Notes

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

## Parents

- Abelian quantum-double stabilizer code — Modular-qudit surface code Hamiltonians admit topological phases associated with \(\mathbb{Z}_q\) topological order [2].
- Generalized homological-product CSS code

## Children

- Kitaev surface code — The modular-qudit surface code for \(q=2\) reduces to the surface code.
- \([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code — The qudit Shor code is a small qudit surface code on a Möbius strip with smooth boundary, which is obtained from removing a face of the tesselation of the projective plane \(\mathbb{R}P^2\) [6; Fig. 4].

## Cousins

- Analog surface code — The analog surface code realizes a straightforward extension of the modular-qudit surface code to infinite local dimension, \(q\to\infty\). There are two types of anyons, \(e\) and \(m\), with each type being valued in \(U(1)\) as opposed to \(\mathbb{Z}_q\) for the qudit surface code.
- Double-semion stabilizer code — The exchange statistics of the anyon for the double-semion code coincides with a subset of anyons in the \(\mathbb{Z}_4\), but the fusion rules are different. The double-semion code can be obtained from the \(\mathbb{Z}_4\) surface code by condensing the anyon \(e^2 m^2\) [7] or by gauging the one-form symmetry associated with said anyon [7; Footnote 20].
- \(\mathbb{Z}_q^{(1)}\) subsystem 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 [7; Sec. 7.3]. During this process, the square lattice is effectively expanded to a hexagonal lattice [7; Fig. 12].

## 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 ZN toric code”, Journal of Mathematical Physics 64, (2023) arXiv:2211.00299 DOI
- [4]
- H. Anwar et al., “Fast decoders for qudit topological codes”, New Journal of Physics 16, 063038 (2014) arXiv:1311.4895 DOI
- [5]
- F. H. E. Watson, H. Anwar, and D. E. Browne, “Fast fault-tolerant decoder for qubit and qudit surface codes”, Physical Review A 92, (2015) arXiv:1411.3028 DOI
- [6]
- M. H. Freedman and D. A. Meyer, “Projective plane and planar quantum codes”, (1998) arXiv:quant-ph/9810055
- [7]
- T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI

## 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