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

- Hopf-algebra quantum-double code — The modular-qudit surface code can be generalized to a Hopf-algebra quantum-double code whose ground states remain the same but whose excitations are based on quasitriangular semisimple Hopf algebras of \(\mathbb{Z}_q\) [7].
- 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.
- Twist-defect surface code — Twist-defect surface codes have been extended to prime-dimensional qudits [8].
- 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 \(e^2 m^2\) [9] or by gauging [10–12,12] the one-form symmetry associated with said anyon [9; 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 [9; Sec. 7.3]. During this process, the square lattice is effectively expanded to a hexagonal lattice [9; 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]
- A. Conlon, D. Pellegrino, and J. K. Slingerland, “Modified toric code models with flux attachment from Hopf algebra gauge theory”, Journal of Physics A: Mathematical and Theoretical 56, 295302 (2023) arXiv:2210.07909 DOI
- [8]
- M. G. Gowda and P. K. Sarvepalli, “Quantum computation with generalized dislocation codes”, Physical Review A 102, (2020) DOI
- [9]
- T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
- [10]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [11]
- L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter”, Journal of High Energy Physics 2017, (2017) arXiv:1605.01640 DOI
- [12]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 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