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Modular-qudit surface code [1–4]

Alternative Names: \(\mathbb{Z}_q\) surface code.

Description

Extension of the surface code to prime-dimensional [1,3] and more general modular qudits. 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 depend on the qudit dimension and the stabilizer generators. In particular, the torus ground-state degeneracy can depend on the lattice periods, and the model can realize either topological order or symmetry-protected topological order [4,5].

More generally, the \(\mathbb{Z}_q\) toric-code Hamiltonian admits rotor limits related to compact \(U(1)\) lattice gauge theory and a continuum coupled-oscillator limit whose ground states coincide with those of the CV surface code [6].

Protection

The code distance is the minimum weight of a non-trivial homology or cohomology class, i.e., the length of the shortest non-contractible cycle in the cellulation or in its dual [2]. Planar discs with holes yield local modular-qudit surface-code families whose logical operators are described by relative homology classes [2].

Gates

A magic-state preparation routine for the \(\mathbb{Z}_4\) surface code traverses through the \(D_4\) quantum double model [7].Gates can be implemented through topological operations corresponding to elements of the mapping class group, which is generated by Dehn-twists along non-contractible cycles [8].

Decoding

Renormalization group decoder [9–11].

Code Capacity Threshold

Generalized bit-flip noise: \(13\%\) for \(\mathbb{Z}_3\) under the renormalization group decoder, with thresholds increasing with qudit dimension for dimensions up to six [9].

Realizations

State preparation, anyon creation, anyon fusion, and transfer of entanglement between anyons and defects in a 24-qutrit trapped-ion device by Quantinuum [12].

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 [13–22] the charge conjugation symmetry of the \(\mathbb{Z}_3\) surface code [23]. A magic-state preparation routine for the \(\mathbb{Z}_4\) surface code traverses through the \(D_4\) quantum double model [7].
  • 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 \(\mathbb{Z}_q\) [24].
  • Compactified \(\mathbb{R}\) gauge theory code— The compactified \(\mathbb{R}\) gauge theory code can be thought of as a realization of the \(q\to\infty\) \(U(1)\) rotor limit [6] of the qudit surface code as a bosonic stabilizer code.
  • Analog surface code— The analog surface code can be thought of as a realization of the \(q\to\infty\) \(\mathbb{R}\) oscillator limit [6] 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 [6] of the qudit surface code as a tiger code.
  • Twist-defect surface code— Twist-defect surface codes have been extended to prime-dimensional qudits [25].
  • Qudit X-cube model code— The qudit X-cube model code is obtained from three orthogonal stacks of modular-qudit surface codes with two qudits per edge, one from each layer containing it [26]. A strong \(X\)-type coupling of each pair condenses composite flux loops and yields the X-cube stabilizers [26]. A field-theoretic description of the qudit X-cube model can be obtained by coupling layers of 2D \(\mathbb{Z}_q\) gauge theory [27].
  • Double-semion stabilizer code— The exchange statistics of the anyons in the double-semion code coincide with those of a subset of the 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 \(e^2 m^2\) [28] or by gauging [13–22] the one-form symmetry associated with said anyon [28; Footnote 20].
  • Six-semion stabilizer code— The six-semion code is obtained from two copies of the \(\mathbb{Z}_4\) surface code by condensing the bosons \(e_1^2 m_1^2\) and \(e_1^2 e_2^2 m_2^2\) [29,30].
  • \(\mathbb{Z}_q^{(1)}\) 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 transversal Clifford gates [28; Sec. 7.3]. During this process, the square lattice is effectively expanded to a honeycomb tiling [28; Fig. 12].

Member of code lists

Primary Hierarchy

Quantum Domain
Modular-qudit Kingdom
Modular-qudit codeQECC Quantum
Modular-qudit USt code
Modular-qudit stabilizer codeStabilizer Hamiltonian-based QECC Quantum
Modular-qudit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum
Parents
Modular-qudit CSS code
Abelian quantum-double stabilizer codeLattice stabilizer Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum
Modular-qudit surface code Hamiltonians admit topological phases associated with \(\mathbb{Z}_q\) topological order [3].
Generalized homological-product CSS codeGeneralized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Modular-qudit surface code
Children
Kitaev surface code
The modular-qudit surface code for \(q=2\) reduces to the surface code.

References

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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.), 2026. https://errorcorrectionzoo.org/c/qudit_surface, arXiv:2606.11484
BibTeX:
@incollection{eczoo_qudit_surface,
title={Modular-qudit surface code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
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.), 2026. https://errorcorrectionzoo.org/c/qudit_surface, arXiv:2606.11484

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