Honeycomb (6.6.6) color code

Honeycomb (6.6.6) color code [1]

Description

2D color code defined on a (typically triangular) patch of the 6.6.6 (honeycomb) tiling. The usual triangular patch has three differently colored boundaries, encodes one logical qubit, and is local-Clifford equivalent to a folded surface/toric code with two smooth and two rough boundaries [2].

Stabilizer generators are shown in Fig. I.

Figure I: Stabilizer generators of the 6.6.6 color code.

Protection

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

Transversal and Permutation-Based Gates

CNOT gate because the code is CSS.Hadamard gates for any qubit geometry which yields a self-dual CSS code.

Gates

Lattice surgery scheme for a hybrid 6.6.6-4.8.8 layout yields lower resource overhead when compared to analogous surface code scheme [5].Low-overhead magic-state distillation circuit using flag qubits [6] or lattice surgery [7].

Decoding

Distance-three measurement schedule based on detector error models [8].Message-passing decoder [9].Adaptation of the restriction decoder [4].Neural-network decoder [10].Möbius matching decoder gives low logical failure rate [11] and has an open-source implementation called Chromöbius [12].AMBP4, a quaternary version [13] of the MBP decoder [14].MaxSAT-based decoder [15].Height-bound decision-tree decoder (DTD) [16].Most likely error (MLE) decoder [17].Neural network decoder [17].Frontier decoder, a pruned dynamic-programming decoder [18].

Fault Tolerance

Fault-tolerant syndrome extraction circuits using flag qubits [4,19].

Code Capacity Threshold

Independent \(X,Z\) noise: \(p_X = 7.8\%\) under message-passing decoder [9], \(8.7\%\) under projection decoder [20], \(\geq 6\%\) under rescaling decoder [21], \(9.0\%\) under Möbius matching decoder [11], \(10.1\%\) under MaxSAT-based decoder [15], \(8.2\%\) under concatenated MWPM decoder [22], \(7.31(5)\%\) under cell-matching decoder [23; Sec. V.E], and close to the optimal value under the Frontier decoder [18]. 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 [24,25], calculated to be \(10.9(2)\%\) in Ref. [25] and \(10.97(1)\%\) in Ref. [26].Depolarizing channel: \(12.6\%\) under the restriction decoder [4] and the projection decoder [20], and \(\approx 14.5\%\) under AMBP4 decoding [13; Fig. 12].

Threshold

The threshold under ML decoding with measurement errors corresponds to the value of a critical point of a three-dimensional disordered Ising model, estimated to be \(4.8(2)\%\) [27].Circuit-level noise: \(0.2\%\) using two flag qubits per stabilizer generator and the restriction decoder [4], and \(0.46\%\) under concatenated MWPM decoder [22].A measurement threshold of one [28].

Realizations

Superconducting qubits: transversal Clifford gates, randomized logical benchmarking, and magic-state injection demonstrated on distance-three and five 6.6.6 color codes on the Willow device by Google Quantum AI [17]. Logical state teleportation using lattice surgery performed between two distance-three color codes. Magic-state cultivation was demonstrated on a device by Google Quantum AI by code switching between a distance-three 6.6.6 color code and distance-five \(XZZX\) surface code and decoding with the Tesseract decoder [29].

Cousins

Honeycomb tiling— The 6.6.6 color code is defined on the honeycomb tiling.
Concatenated GKP code— GKP codes have been concatenated with the 6.6.6 color code [30].
Tensor-network code— Larger 6.6.6 color codes can be constructed by contracting legs of tensors of smaller codes [31; Fig. 5].
2D DA color code— The parent topological phase of the 2D DA color code is realized by two copies of the 6.6.6 color code [32; Sec. III.A].
Ruby Floquet code— One third of the time during the XYZ ruby measurement schedule, the ISG is that of the 6.6.6 color code concatenated with a three-qubit repetition code.
Bivariate bicycle (BB) code— Certain bivariate bicycle codes are equivalent to a family of 6.6.6 color codes [33; Exam. 1]. The 6.6.6 color code on a torus is the self-dual BB code with \(A=1+x+y\) [34; Exam. 2].
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 an analogous surface code scheme [5].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeQubit Stabilizer Hamiltonian-based QECC Quantum

Qubit QLDPC codeStabilizer Hamiltonian-based QECC Quantum

Color codeQubit CSS Stabilizer Hamiltonian-based QECC Quantum

2D color codeTwist-defect color QLDPC Qubit Generalized homological-product Lattice stabilizer CSS Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum

Parents

2D color code

Lifted-product (LP) codeGeneralized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum

The 6.6.6 color code can be formulated directly as an LP code [35].

Honeycomb (6.6.6) color code

Children

\([[18,4,4]]\) 6.6.6 color code

The \([[18,4,4]]\) code is the \(\ell=m=3\) member of the bivariate bicycle family \(A=1+x+xy\), \(B=1+y+xy\), which yields 6.6.6 color codes whenever \(\ell\) and \(m\) are multiples of three [33; Exam. 1].

\([[4,2,2]]\) Four-qubit code

The \([[4,2,2]]\) code can be interpreted as a 2D color code on a trapezoidal patch that makes up two-thirds of a hexagon of the 6.6.6 tiling [36,37].

\([[6,4,2]]\) error-detecting code

The \([[6,4,2]]\) error-detecting code is a color code defined on a single hexagon of the 6.6.6 or 4.6.12 tilings. The \([[6,4,2]]\) code can be concatenated with the surface code to yield the 6.6.6 color code [38; 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.

XYZ color code

The XYZ color code is a 6.6.6 color code presented in non-CSS form. It is obtained by applying a single-qubit Clifford rotation to every qubit on one sublattice of the honeycomb [39; Sec. 2.1]. This rotation maps the \(X\)- and \(Z\)-type generators to \(XZXZXZ\) and \(ZYZYZY\), respectively. The domain wall color code is likewise obtained by applying Hadamard gates to half of the qubits [40]. Since both codes are defined by such rotations, every instance is equivalent to a CSS 6.6.6 color code.

References

[1]
H. Bombin and M. A. Martin-Delgado, “Topological Quantum Distillation”, Physical Review Letters 97, (2006) arXiv:quant-ph/0605138 DOI
[2]
A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
[3]
A. J. Landahl, J. T. Anderson, and P. R. Rice, “Fault-tolerant quantum computing with color codes”, (2011) arXiv:1108.5738
[4]
C. Chamberland, A. Kubica, T. J. Yoder, and G. Zhu, “Triangular color codes on trivalent graphs with flag qubits”, New Journal of Physics 22, 023019 (2020) arXiv:1911.00355 DOI
[5]
F. Thomsen, M. S. Kesselring, S. D. Bartlett, and B. J. Brown, “Low-overhead quantum computing with the color code”, Physical Review Research 6, (2024) arXiv:2201.07806 DOI
[6]
C. Chamberland and K. Noh, “Very low overhead fault-tolerant magic state preparation using redundant ancilla encoding and flag qubits”, npj Quantum Information 6, (2020) arXiv:2003.03049 DOI
[7]
S.-H. Lee, F. Thomsen, N. Fazio, B. J. Brown, and S. D. Bartlett, “Low-Overhead Magic State Distillation with Color Codes”, PRX Quantum 6, (2025) arXiv:2409.07707 DOI
[8]
P.-J. H. S. Derks, A. Townsend-Teague, A. G. Burchards, and J. Eisert, “Designing fault-tolerant circuits using detector error models”, Quantum 9, 1905 (2025) arXiv:2407.13826 DOI
[9]
P. Sarvepalli and R. Raussendorf, “Efficient decoding of topological color codes”, Physical Review A 85, (2012) arXiv:1111.0831 DOI
[10]
N. Maskara, A. Kubica, and T. Jochym-O’Connor, “Advantages of versatile neural-network decoding for topological codes”, Physical Review A 99, (2019) arXiv:1802.08680 DOI
[11]
K. Sahay and B. J. Brown, “Decoder for the Triangular Color Code by Matching on a Möbius Strip”, PRX Quantum 3, (2022) arXiv:2108.11395 DOI
[12]
C. Gidney and C. Jones, “New circuits and an open source decoder for the color code”, (2023) arXiv:2312.08813
[13]
K.-Y. Kuo and C.-Y. Lai, “Comparison of 2D topological codes and their decoding performances”, 2022 IEEE International Symposium on Information Theory (ISIT) 186 (2022) arXiv:2202.06612 DOI
[14]
K.-Y. Kuo and C.-Y. Lai, “Exploiting degeneracy in belief propagation decoding of quantum codes”, npj Quantum Information 8, (2022) arXiv:2104.13659 DOI
[15]
L. Berent, L. Burgholzer, P.-J. H. S. Derks, J. Eisert, and R. Wille, “Decoding quantum color codes with MaxSAT”, Quantum 8, 1506 (2024) arXiv:2303.14237 DOI
[16]
K. R. Ott, B. Hetényi, and M. E. Beverland, “Decision-tree decoders for general quantum LDPC codes”, (2025) arXiv:2502.16408
[17]
N. Lacroix et al., “Scaling and logic in the colour code on a superconducting quantum processor”, Nature 645, 614 (2025) arXiv:2412.14256 DOI
[18]
A. Leverrier and R. Urbanke, “Approximating optimal decoding of quantum LDPC codes with narrow frontiers”, (2026) arXiv:2606.20513
[19]
C. Chamberland and M. E. Beverland, “Flag fault-tolerant error correction with arbitrary distance codes”, Quantum 2, 53 (2018) arXiv:1708.02246 DOI
[20]
N. Delfosse, “Decoding color codes by projection onto surface codes”, Physical Review A 89, (2014) arXiv:1308.6207 DOI
[21]
P. Parrado-Rodríguez, M. Rispler, and M. Müller, “Rescaling decoder for two-dimensional topological quantum color codes on 4.8.8 lattices”, Physical Review A 106, (2022) arXiv:2112.09584 DOI
[22]
S.-H. Lee, A. Li, and S. D. Bartlett, “Color code decoder with improved scaling for correcting circuit-level noise”, Quantum 9, 1609 (2025) arXiv:2404.07482 DOI
[23]
S. J. S. Tan, I. Gill, E. Huang, P. Liu, C. Zhao, H. Dehghani, A. Kubica, H. Zhou, and A. Dua, “Generalized matching decoders for 2D topological translationally-invariant codes”, (2026) arXiv:2603.05402
[24]
H. Nishimori, “Geometry-Induced Phase Transition in the ±JIsing Model”, Journal of the Physical Society of Japan 55, 3305 (1986) DOI
[25]
H. G. Katzgraber, H. Bombin, and M. A. Martin-Delgado, “Error Threshold for Color Codes and Random Three-Body Ising Models”, Physical Review Letters 103, (2009) arXiv:0902.4845 DOI
[26]
M. Ohzeki, “Accuracy thresholds of topological color codes on the hexagonal and square-octagonal lattices”, Physical Review E 80, (2009) arXiv:0903.2102 DOI
[27]
R. S. Andrist, H. G. Katzgraber, H. Bombin, and M. A. Martin-Delgado, “Tricolored lattice gauge theory with randomness: fault tolerance in topological color codes”, New Journal of Physics 13, 083006 (2011) arXiv:1005.0777 DOI
[28]
D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
[29]
E. Rosenfeld et al., “Magic state cultivation on a superconducting quantum processor”, (2025) arXiv:2512.13908
[30]
M. Lin and K. Noh, “Exploring the quantum capacity of a Gaussian random-displacement channel using Gottesman-Kitaev-Preskill codes and maximum-likelihood decoding”, Physical Review A 111, (2025) arXiv:2411.04277 DOI
[31]
T. Farrelly, D. K. Tuckett, and T. M. Stace, “Local tensor-network codes”, New Journal of Physics 24, 043015 (2022) arXiv:2109.11996 DOI
[32]
M. Davydova, N. Tantivasadakarn, S. Balasubramanian, and D. Aasen, “Quantum computation from dynamic automorphism codes”, Quantum 8, 1448 (2024) arXiv:2307.10353 DOI
[33]
J. N. Eberhardt, F. R. F. Pereira, and V. Steffan, “Pruning qLDPC codes: Towards bivariate bicycle codes with open boundary conditions”, (2024) arXiv:2412.04181
[34]
Z. Liang and Y.-A. Chen, “Self-dual bivariate bicycle codes with transversal Clifford gates”, (2026) arXiv:2510.05211
[35]
Y. Tan, B. Roberts, N. Tantivasadakarn, B. Yoshida, and N. Y. Yao, “Fracton models from product codes”, Physical Review Research 7, (2025) arXiv:2312.08462 DOI
[36]
M. S. Kesselring, J. C. Magdalena de la Fuente, F. Thomsen, J. Eisert, S. D. Bartlett, and B. J. Brown, “Anyon Condensation and the Color Code”, PRX Quantum 5, (2024) arXiv:2212.00042 DOI
[37]
R. S. Gupta et al., “Encoding a magic state with beyond break-even fidelity”, Nature 625, 259 (2024) arXiv:2305.13581 DOI
[38]
B. Criger and B. Terhal, “Noise thresholds for the [4,2,2]-concatenated toric code”, Quantum Information and Computation 16, 1261 (2016) arXiv:1604.04062 DOI
[39]
J. F. S. Miguel, D. J. Williamson, and B. J. Brown, “A cellular automaton decoder for a noise-bias tailored color code”, Quantum 7, 940 (2023) arXiv:2203.16534 DOI
[40]
K. Tiurev, A. Pesah, P.-J. H. S. Derks, J. Roffe, J. Eisert, M. S. Kesselring, and J.-M. Reiner, “Domain Wall Color Code”, Physical Review Letters 133, (2024) arXiv:2307.00054 DOI

Page edit log

Victor V. Albert (2026-11-08) — most recent
Victor V. Albert (2026-10-27)
Anthony Leverrier (2026-07-22)
Victor V. Albert (2026-06-08)
Eric Huang (2024-03-18)
Victor V. Albert (2023-11-13)
Cella Kove (2023-07-24)

Cite as:

“Honeycomb (6.6.6) color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/triangular_color, arXiv:2606.11484

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