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Modular-qudit surface code[13]

Alternative names: \(\mathbb{Z}_q\) 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.

Gates

A magic-state preparation routine for the \(\mathbb{Z}_4\) surface code traverses through the \(D_4\) quantum double model [4].

Decoding

Renormalization group decoder [5,6].

Realizations

State preparation, anyon creation, anyon fusion, and transfer of entanglement between anyons and defect in a 24 qubit trapped ion device by Quantinuum [7].

Notes

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

Cousins

  • Dihedral \(G=D_m\) quantum-double code— The \(D_3\) quantum double model can be obtained by gauging [817] the charge conjugation symmetry of the \(\mathbb{Z}_3\) surface code [18]. A magic-state preparation routine for the \(\mathbb{Z}_4\) surface code traverses through the \(D_4\) quantum double model [4].
  • 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 \(\mathbb{Z}_q\) [19].
  • Compactified \(\mathbb{R}\) gauge theory code— The tiger surface code can be thought of as a realization of the \(q\to\infty\) \(U(1)\) rotor limit [20] of the qudit surface code as a bosonic stabilizer code.
  • Analog surface code— The tiger surface code can be thought of as a realization of the \(q\to\infty\) \(\mathbb{R}\) oscillator limit [20] of the qudit surface code as a bosonic stabilizer code.
  • Tiger surface code— The tiger surface code can be thought of as a realization of the \(q\to\infty\) \(U(1)\) rotor limit [20] of the qudit surface code as a tiger code.
  • Twist-defect surface code— Twist-defect surface codes have been extended to prime-dimensional qudits [21].
  • Qudit X-cube model code— A field-theoretic description of the qudit X-cube model can be obtained by coupling layers of 2D \(\mathbb{Z}_q\) gauge theory [22].
  • Double-semion stabilizer code— The exchange statistics of the anyon for the double-semion code coincides with a subset of anyons in the \(\mathbb{Z}_4\) surface code, but the fusion rules are different. The double-semion code can be obtained from the \(\mathbb{Z}_4\) surface code by condensing the anyon \(e^2 m^2\) [23] or by gauging [817] the one-form symmetry associated with said anyon [23; 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 [23; Sec. 7.3]. During this process, the square lattice is effectively expanded to a honeycomb tiling [23; Fig. 12].

Primary Hierarchy

Parents
Modular-qudit surface code Hamiltonians admit topological phases associated with \(\mathbb{Z}_q\) topological order [2].
Modular-qudit surface code
Children
The modular-qudit surface code for \(q=2\) reduces to the surface 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 \(\mathbb{R}P^2\) [24; Fig. 4].

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]
S.-J. Huang and Y. Chen, “Generating logical magic states with the aid of non-Abelian topological order”, (2025) arXiv:2502.00998
[5]
H. Anwar, B. J. Brown, E. T. Campbell, and D. E. Browne, “Fast decoders for qudit topological codes”, New Journal of Physics 16, 063038 (2014) arXiv:1311.4895 DOI
[6]
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
[7]
M. Iqbal et al., “Qutrit Toric Code and Parafermions in Trapped Ions”, (2024) arXiv:2411.04185
[8]
M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
[9]
J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac, and F. Verstraete, “Gauging Quantum States: From Global to Local Symmetries in Many-Body Systems”, Physical Review X 5, (2015) arXiv:1407.1025 DOI
[10]
S. Vijay, J. Haah, and L. Fu, “Fracton topological order, generalized lattice gauge theory, and duality”, Physical Review B 94, (2016) arXiv:1603.04442 DOI
[11]
D. J. Williamson, “Fractal symmetries: Ungauging the cubic code”, Physical Review B 94, (2016) arXiv:1603.05182 DOI
[12]
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
[13]
A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
[14]
W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
[15]
K. Dolev, V. Calvera, S. S. Cree, and D. J. Williamson, “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022) arXiv:2108.11402 DOI
[16]
T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes I. Gauging and dualities”, (2023) arXiv:2310.16032
[17]
D. J. Williamson and T. J. Yoder, “Low-overhead fault-tolerant quantum computation by gauging logical operators”, (2024) arXiv:2410.02213
[18]
A. Lyons, C. F. B. Lo, N. Tantivasadakarn, A. Vishwanath, and R. Verresen, “Protocols for Creating Anyons and Defects via Gauging”, (2025) arXiv:2411.04181
[19]
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
[20]
V. V. Albert, S. Pascazio, and M. H. Devoret, “General phase spaces: from discrete variables to rotor and continuum limits”, Journal of Physics A: Mathematical and Theoretical 50, 504002 (2017) arXiv:1709.04460 DOI
[21]
M. G. Gowda and P. K. Sarvepalli, “Quantum computation with generalized dislocation codes”, Physical Review A 102, (2020) DOI
[22]
P. Gorantla, A. Prem, N. Tantivasadakarn, and D. J. Williamson, “String-Membrane-Nets from Higher-Form Gauging: An Alternate Route to \(p\)-String Condensation”, (2025) arXiv:2505.13604
[23]
T. D. Ellison, Y.-A. Chen, A. Dua, W. Shirley, N. Tantivasadakarn, and D. J. Williamson, “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
[24]
M. H. Freedman and D. A. Meyer, “Projective plane and planar quantum codes”, (1998) arXiv:quant-ph/9810055
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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. https://errorcorrectionzoo.org/c/qudit_surface
BibTeX:
@incollection{eczoo_qudit_surface, title={Modular-qudit surface code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qudit_surface} }
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“Modular-qudit surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudit_surface

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits/stabilizer/topological/qudit_surface.yml.