Modular-qudit surface code[13] 


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.


Renormalization-group decoder [4].


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




  • 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.
  • \(\mathbb{Z}_q^{(1)}\) 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 [6; Sec. 7.3]. During this process, the square lattice is effectively expanded to a hexagonal lattice [6; Fig. 12].
  • 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 one-form symmetry associated with said anyon [6; Footnote 18].


A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003) arXiv:quant-ph/9707021 DOI
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
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
H. Anwar et al., “Fast decoders for qudit topological codes”, New Journal of Physics 16, 063038 (2014) arXiv:1311.4895 DOI
M. H. Freedman and D. A. Meyer, “Projective plane and planar quantum codes”, (1998) arXiv:quant-ph/9810055
T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, (2022) arXiv:2211.03798
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Zoo Code ID: qudit_surface

Cite as:
“Modular-qudit surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
  title={Modular-qudit surface code},
  booktitle={The Error Correction Zoo},
  editor={Albert, Victor V. and Faist, Philippe},
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“Modular-qudit surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.