Modular-qudit 3D surface code

Modular-qudit 3D surface code [1,2]

Description

A generalization of the 3D surface code to modular qudits. Qudits are placed on edges, \(Z\)-type stabilizer generators are placed on square plaquettes oriented in all three directions, and \(X\)-type stabilizers are placed on the six edges neighboring every vertex [1].

Cousin

Qudit X-cube model code— Three orthogonal stacks of modular-qudit surface codes with two qudits per edge, one from each layer containing it, yield the modular-qudit 3D surface code when composite charges condense [3]. The same layers yield the qudit X-cube model code when composite flux loops condense. With only the \(X\)-type coupling, the transition from decoupled layers into the qudit X-cube model phase is dual to flux-loop condensation out of the modular-qudit 3D surface-code phase [3].

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

Quantum-triple codeTopological Hamiltonian-based QECC Quantum

A quantum triple model for the group \(G=\mathbb{Z}_q\) is a modular-qudit 3D surface code.

Generalized homological-product CSS codeGeneralized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum

3D lattice stabilizer codeLattice stabilizer Stabilizer Hamiltonian-based QECC Quantum

Abelian topological codeTopological Hamiltonian-based QECC Quantum

The modular-qudit 3D surface code realizes 3D \(\mathbb{Z}_q\) gauge theory with bosonic charge and loop excitations (BcBl).

Modular-qudit 3D surface code

Children

3D surface code

The qudit 3D surface code reduces to the 3D surface code for \(q=2\). The 3D surface code realizes 3D \(\mathbb{Z}_2\) gauge theory with bosonic charge and loop excitations (BcBl). The welded surface code does not satisfy homogeneous topological order [4].

References

[1]
H. Moradi and X.-G. Wen, “Universal topological data for gapped quantum liquids in three dimensions and fusion algebra for non-Abelian string excitations”, Physical Review B 91, (2015) arXiv:1404.4618 DOI
[2]
C. Lee, Y. Hu, G. Y. Cho, and H. Watanabe, “<mml:math xmlns:mml=”http://www.w3.org/1998/Math/MathML”> <mml:msub> <mml:mi mathvariant=”double-struck”>Z</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> generalizations of three-dimensional stabilizer codes”, Physical Review B 112, (2025) arXiv:2504.09847 DOI
[3]
S. Vijay, “Isotropic Layer Construction and Phase Diagram for Fracton Topological Phases”, (2017) arXiv:1701.00762
[4]
J. Haah, “A degeneracy bound for homogeneous topological order”, SciPost Physics 10, (2021) arXiv:2009.13551 DOI

Page edit log

Victor V. Albert (2026-06-08) — most recent
Victor V. Albert (2024-04-20)

Cite as:

“Modular-qudit 3D surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/qudit_3d_surface, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits/stabilizer/topological/qudit_3d_surface.yml.