Hexagonal \(CZ\) code

Hexagonal \(CZ\) code[1]

Description

A cubic theory code defined on a hexagonal lattice in 2D. Its stabilizers are products of Pauli-\(Z\) operators and \(CZ\) gates [1; Fig. 6][2; Fig. 3]. The ground-state subspace of the hexagonal \(CZ\) code realizes the topological order of the \(G=\mathbb{Z}^3_2\) TQD model [1,3], which is the same topological order as the \(G=D_4\) quantum double [4,5]. The stabilizers include \(CZ\) operators acting on hexagonal loops, but a reduced version exists where only two \(CZ\) gates act on each loop [2].

Gates

The hexagonal \(CZ\) code can be obtained from two surface codes by gauging [6–15] their logical \(CZ\) gate [2]. Gates on the two surface codes in the third level of the Clifford hierarchy, such as \(CZ\) gates, can be realized fault tolerantly by performing this procedure and reversing it [2].

Fault Tolerance

The hexagonal \(CZ\) code can be obtained from two surface codes by gauging [6–15] their logical \(CZ\) gate [2]. Gates on the two surface codes in the third level of the Clifford hierarchy, such as \(CZ\) gates, can be realized fault tolerantly by performing this procedure and reversing it [2].

Realizations

Signatures of the phase detected in a 27-qubit trapped-ion device by Quantinuum [16]. Preparation of ground states and braiding of anyons has also been performed.

Notes

Popular summary of realization of non-Abelian topological order in Quanta Magazine.

Cousins

Dihedral \(G=D_m\) quantum-double code— The ground-state subspace of the hexagonal \(CZ\) code realizes the topological order of the \(G=\mathbb{Z}^3_2\) TQD model [1,3], which is the same topological order as the \(G=D_4\) quantum double [4,5].
Kitaev surface code— The hexagonal \(CZ\) code can be obtained from two surface codes by gauging [6–15] their logical \(CZ\) gate [2]. Gates on the two surface codes in the third level of the Clifford hierarchy, such as \(CZ\) gates, can be realized fault tolerantly by performing this procedure and reversing it [2].
Symmetry-protected topological (SPT) code— The hexagonal \(CZ\) code can be obtained by gauging [6–15] the symmetry of a particular SPT [1].
Brickwork \(XS\) stabilizer code— The brickwork \(XS\) stabilizer code and the hexagonal \(CZ\) code realize the same topological phases and are equivalent via a local unitary [2,17].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Cubic theory codeTopological Hamiltonian-based QECC Quantum

Parents

Cubic theory code

The cubic theory code defined on a hexagonal lattice reduces to the hexagonal \(CZ\) code [3].

Abelian TQD codeAbelian topological Topological Hamiltonian-based QECC Quantum

The ground-state subspace of the hexagonal \(CZ\) code realizes the topological order of the \(G=\mathbb{Z}^3_2\) TQD model [1,3], which is the same topological order as the \(G=D_4\) quantum double [4,5].

Hexagonal \(CZ\) code

References

[1]: B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI
[2]: M. Davydova, A. Bauer, J. C. M. de la Fuente, M. Webster, D. J. Williamson, and B. J. Brown, “Universal fault tolerant quantum computation in 2D without getting tied in knots”, (2025) arXiv:2503.15751
[3]: P.-S. Hsin, R. Kobayashi, and G. Zhu, “Non-Abelian Self-Correcting Quantum Memory”, (2024) arXiv:2405.11719
[4]: M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
[5]: L. Lootens, B. Vancraeynest-De Cuiper, N. Schuch, and F. Verstraete, “Mapping between Morita-equivalent string-net states with a constant depth quantum circuit”, Physical Review B 105, (2022) arXiv:2112.12757 DOI
[6]: M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
[7]: 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
[8]: 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
[9]: D. J. Williamson, “Fractal symmetries: Ungauging the cubic code”, Physical Review B 94, (2016) arXiv:1603.05182 DOI
[10]: 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
[11]: A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
[12]: W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
[13]: 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
[14]: T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes I. Gauging and dualities”, (2023) arXiv:2310.16032
[15]: D. J. Williamson and T. J. Yoder, “Low-overhead fault-tolerant quantum computation by gauging logical operators”, (2024) arXiv:2410.02213
[16]: M. Iqbal et al., “Non-Abelian topological order and anyons on a trapped-ion processor”, Nature 626, 505 (2024) arXiv:2305.03766 DOI
[17]: Benjamin J. Brown, private communication, 2025

Page edit log

Victor V. Albert (2025-07-03) — most recent
Benjamin J. Brown (2025-07-03)

Cite as:

“Hexagonal \(CZ\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/hexagonal_cz

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/nonstabilizer/cz_stabilizer/hexagonal_cz.yml.