Magic Boundaries of 3D Color Codes

Zijian Song; Guanyu Zhu

doi:10.22331/q-2025-08-27-1831

Magic Boundaries of 3D Color Codes

Zijian Song¹ and Guanyu Zhu²

¹Department of Physics and Astronomy, University of California, Davis, California 95616, USA
²IBM Quantum, T.J. Watson Research Center, Yorktown Heights, NY 10598 USA

Published:	2025-08-27, volume 9, page 1831
Editor:	Alioscia Hamma
Eprint:	arXiv:2404.05033v4
Doi:	https://doi.org/10.22331/q-2025-08-27-1831
Citation:	Quantum 9, 1831 (2025).

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Abstract

We investigate boundaries of 3D color codes and provide a systematic classification into 101 distinct boundary types, including two novel classes. The first class consists of 1 boundary and is generated by sweeping the codimension-1 (2D) $T$-domain wall across the system and attaching it to the $X$-boundary that condenses only magnetic fluxes. Since the $T$-domain wall cannot condense on the $X$-boundary, a new $\textit{magic boundary}$ is produced, where the boundary stabilizers contain $XS$-stabilizers going beyond the conventional Pauli stabilizer formalism, and hence contains 'magic'. Neither electric nor magnetic excitations can condense on such a magic boundary, and only the composite of the magnetic flux and codimension-2 (1D) $S$-domain wall can condense on it, which makes the magic boundary going beyond the classification of the Lagrangian subgroup. The second class consists of 70 boundaries and is generated by sweeping the $S$-domain wall across a codimension-1 submanifold and attaching it to the boundary. This generates a codimension-2 (1D) $\textit{nested boundary}$ at the intersection. We also connect these novel boundaries to their previously discovered counterpart in the $\mathbb{Z}_2^3$ gauge theory, equivalent to three copies of 3D toric codes, where the $S$ and $T$ domain walls correspond to gauged symmetry-protected topological (SPT) defects. New boundaries are produced whenever the corresponding symmetry of the SPT defect remains unbroken on the boundary. Applications of the magic boundaries include implementing fault-tolerant non-Clifford logical gates, e.g., in the context of fractal topological codes.

Featured image: The magnetic loop accompanied by the codimension-1 $S$-domain wall can condense on the magic boundary, whereas the magnetic loop by itself cannot.

Popular summary

The paper “Magic Boundaries of 3D Color Codes” classifies the boundaries of a quantum error-correcting system known as the 3D color code into 101 types. One, called the magic boundary, has the exotic property that no topological excitations from the bulk can condense on it. This exotic feature could be harnessed for universal quantum computing. The findings connect the geometry of these codes with powerful computational capabilities and could inspire new pathways for developing more efficient and robust quantum codes.

► BibTeX data

@article{Song2025magicboundariesofd,
  doi = {10.22331/q-2025-08-27-1831},
  url = {https://doi.org/10.22331/q-2025-08-27-1831},
  title = {Magic {B}oundaries of 3{D} {C}olor {C}odes},
  author = {Song, Zijian and Zhu, Guanyu},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1831},
  month = aug,
  year = {2025}
}

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[2] Gabrielle Tournaire, Marvin Schwiering, Robert Raussendorf, and Sven Bachmann, "A 3D lattice defect and efficient computations in topological MBQC", Quantum 10, 1997 (2026).

[3] Andreas Bauer and Julio C. Magdalena de la Fuente, "Planar Fault-Tolerant Circuits for Non-Clifford Gates on the 2D Color Code", PRX Quantum 7 2, 020367 (2026).

[4] Rui Wen, "String condensation and topological holography for 2+1D gapless SPT", arXiv:2408.05801, (2024).

[5] Po-Shen Hsin, Ryohei Kobayashi, and Guanyu Zhu, "Classifying Logical Gates in Quantum Codes via Cohomology Operations and Symmetry", arXiv:2411.15848, (2024).

[6] Victor V. Albert and Philippe Faist, "Handbook of Error-Correcting Codes", arXiv:2606.11484, (2026).

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