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].
Primary Hierarchy
Parents
The cubic theory code defined on a hexagonal lattice reduces to the hexagonal \(CZ\) code [3].
Hexagonal \(CZ\) code
References
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- M. Iqbal et al., “Non-Abelian topological order and anyons on a trapped-ion processor”, Nature 626, 505 (2024) arXiv:2305.03766 DOI
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- 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