Honeycomb (6.6.6) color code[1] 


Triangular color code defined on a patch of the 6.6.6 (honeycomb) tiling.

Stabilizer generators are shown in Figure I.

Figure I: Stabilizer generators of the 6.6.6 color code.


There is a \([[(3d^2+1)/4, 1, d]]\) code family [2; Fig. 2] and a \([[(3d-1)^2/4, 1, d]]\) code family [3].

Transversal Gates

CNOT gate because the code is CSS.Hadamard gates for any qubit geometry which yields a self-dual CSS code.Transversal \(S\) gate [1,2].


Lattice surgery scheme for a hybrid 6.6.6-4.8.8 layout yields lower resource overhead when compared to analogous surface code scheme [4].Low-overhead magic-state distillation circuit using flag qubits [5].


Message-passing decoder [6].Adaptation of the restriction decoder [3].Neural-network decoder [7].Mobius matching decoder gives low logical failure rate [8] and has an open-source implementation called Chromobius [9].AMBP4, a quaternary version [10] of the MBP decoder [11].MaxSAT-based decoder [12].

Fault Tolerance

Fault-tolerant syndrome extraction circuits using flag qubits [3,13].

Code Capacity Threshold

Independent \(X,Z\) noise: \(p_X = 7.8\%\) under message-passing decoder [6], \(8.7\%\) under projection decoder [14], \(\geq 6\%\) under rescaling decoder [15], \(9.0\%\) under Mobius matching decoder [8], \(10.1\%\) under MaxSAT-based decoder [12], and \(8.2\%\) under concatenated MWPM decoder [16]. The threshold under ML decoding corresponds to the value of a critical point of the two-dimensional three-body random-bond Ising model (RBIM) on the Nishimori line [17,18], calculated to be \(10.9(2)\%\) in Ref. [18] and \(10.97(1)\%\) in Ref. [19].Depolarizing channel: \(12.6\%\) under the restriction decoder [3] and the projection decoder [14], and \(\sim 14.5\%\) under AMBP4 decoding [10; Fig. 12].


Circuit-level noise: \(0.2\%\) using two flag qubits per stabilizer generator and the restriction decoder [3], and \(0.46\%\) under concatenated MWPM decoder [16].A measurement threshold of one [20].



  • \([[6,4,2]]\) error-detecting code — The \([[6,4,2]]\) error-detecting code is a color code defined on a single hexagon of the 6.6.6 tiling. The \([[6,4,2]]\) code can be concatenated with the surface code to yield the 6.6.6 color code [21; Appx. A].
  • \([[7,1,3]]\) Steane code — Steane code is a 2D color code defined on a seven-qubit patch of the 6.6.6 tiling.


  • \(A_2\) hexagonal lattice code — The triangular color code is defined on a trivalent lattice such as the honeycomb lattice.
  • Dynamical automorphism (DA) code — The parent topological phase of the 2D DA color code is realized by two copies of the 6.6.6 color code [22].
  • Square-octagon (4.8.8) color code — Lattice surgery scheme for a hybrid 6.6.6-4.8.8 layout yields lower resource overhead when compared to analogous surface code scheme [4].
  • XYZ color code — The XYZ color code is obtained from the 6.6.6 color code by applying single-qubit Clifford rotations on a subset of qubits such that the \(X\)- and \(Z\)-type generators are mapped to \(XZXZXZ\) and \(ZYZYZY\), respectively.


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Zoo Code ID: triangular_color

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“Honeycomb (6.6.6) color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/triangular_color
@incollection{eczoo_triangular_color, title={Honeycomb (6.6.6) color code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/triangular_color} }
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“Honeycomb (6.6.6) color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/triangular_color

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/color/2d_color/triangular_color/triangular_color.yml.