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].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
- Quantum-double code— 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 \(\mathbb{Z}_q\) [8].
- 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 [9] 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 [9] 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 [9] of the qudit surface code as a tiger code.
- Twist-defect surface code— Twist-defect surface codes have been extended to prime-dimensional qudits [10].
- 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 \(e^2 m^2\) [11] or by gauging [12–14,14] the one-form symmetry associated with said anyon [11; 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 [11; Sec. 7.3]. During this process, the square lattice is effectively expanded to a honeycomb tiling [11; Fig. 12].
Primary Hierarchy
Abelian TQD stabilizer codeLattice stabilizer QLDPC Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum
Parents
Modular-qudit surface code Hamiltonians admit topological phases associated with \(\mathbb{Z}_q\) topological order [2].
Generalized homological-product CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
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 \(\mathbb{R}P^2\) [15; Fig. 4].
References
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- 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
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- 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
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- 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
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- M. Iqbal et al., “Qutrit Toric Code and Parafermions in Trapped Ions”, (2024) arXiv:2411.04185
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- 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
- [9]
- 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
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- M. G. Gowda and P. K. Sarvepalli, “Quantum computation with generalized dislocation codes”, Physical Review A 102, (2020) DOI
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- 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
- [12]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [13]
- 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
- [14]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
- [15]
- M. H. Freedman and D. A. Meyer, “Projective plane and planar quantum codes”, (1998) arXiv:quant-ph/9810055
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