Honeycomb (6.6.6) color code

[Jump to code hierarchy]

Honeycomb (6.6.6) color code[1]

Description

2D 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.

Protection

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].

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 [4].Low-overhead magic-state distillation circuit using flag qubits [5] or lattice surgery [6].

Decoding

Distance-three measurement schedule based on detector error models [7].Message-passing decoder [8].Adaptation of the restriction decoder [3].Neural-network decoder [9].Möbius matching decoder gives low logical failure rate [10] and has an open-source implementation called Chromöbius [11].AMBP4, a quaternary version [12] of the MBP decoder [13].MaxSAT-based decoder [14].Most likely error (MLE) decoder [15].Neural network decoder [15].

Fault Tolerance

Fault-tolerant syndrome extraction circuits using flag qubits [3,16].

Code Capacity Threshold

Independent \(X,Z\) noise: \(p_X = 7.8\%\) under message-passing decoder [8], \(8.7\%\) under projection decoder [17], \(\geq 6\%\) under rescaling decoder [18], \(9.0\%\) under Möbius matching decoder [10], \(10.1\%\) under MaxSAT-based decoder [14], and \(8.2\%\) under concatenated MWPM decoder [19]. 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 [20,21], calculated to be \(10.9(2)\%\) in Ref. [21] and \(10.97(1)\%\) in Ref. [22].Depolarizing channel: \(12.6\%\) under the restriction decoder [3] and the projection decoder [17], and \(\approx 14.5\%\) under AMBP4 decoding [12; 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 \(4.8(2)\%\) [23].Circuit-level noise: \(0.2\%\) using two flag qubits per stabilizer generator and the restriction decoder [3], and \(0.46\%\) under concatenated MWPM decoder [19].A measurement threshold of one [24].

Realizations

Superconducting qubits: transversal Clifford gates, randomized logical benchmarking, and magic-state injection demonstrated on distance-three and five triangular color codes on the Willow device by Google Quantum AI [15]. Logical state teleportation using lattice surgery performed between two distance-three color codes.

Cousins

Honeycomb tiling— The triangular color code is defined on the honeycomb tiling.
Concatenated GKP code— GKP codes have been concatenated with the 6.6.6 color code [25].
Tensor-network code— Larger triangular color codes can be constructed by contracting legs of tensors of smaller codes [26; Fig. 5].
XYZ ruby Floquet code— One third of the time during its measurement schedule, the ISG of the XYZ ruby Floquet code is that of the 6.6.6 color code concatenated with a three-qubit repetition code.
Dynamical code— The parent topological phase of the 2D DA color code is realized by two copies of the 6.6.6 color code [27].
Bivariate bicycle (BB) code— Certain bivariate bicycle codes are equivalent to a family of 6.6.6 color codes [28].
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.

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

Quantum rainbow code

Quantum pin codeStabilizer Hamiltonian-based Qubit QECC Quantum

Color code

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

Parents

2D color code

Honeycomb (6.6.6) color code

Children

\([[6,4,2]]\) 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 [29; 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.

References

[1]: H. Bombin and M. A. Martin-Delgado, “Topological Quantum Distillation”, Physical Review Letters 97, (2006) arXiv:quant-ph/0605138 DOI
[2]: A. J. Landahl, J. T. Anderson, and P. R. Rice, “Fault-tolerant quantum computing with color codes”, (2011) arXiv:1108.5738
[3]: 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
[4]: 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
[5]: 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
[6]: S.-H. Lee, F. Thomsen, N. Fazio, B. J. Brown, and S. D. Bartlett, “Low-overhead magic state distillation with color codes”, (2025) arXiv:2409.07707
[7]: P.-J. H. S. Derks, A. Townsend-Teague, A. G. Burchards, and J. Eisert, “Designing fault-tolerant circuits using detector error models”, (2024) arXiv:2407.13826
[8]: P. Sarvepalli and R. Raussendorf, “Efficient decoding of topological color codes”, Physical Review A 85, (2012) arXiv:1111.0831 DOI
[9]: 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
[10]: 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
[11]: C. Gidney and C. Jones, “New circuits and an open source decoder for the color code”, (2023) arXiv:2312.08813
[12]: 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
[13]: 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
[14]: 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
[15]: N. Lacroix et al., “Scaling and logic in the color code on a superconducting quantum processor”, (2024) arXiv:2412.14256
[16]: C. Chamberland and M. E. Beverland, “Flag fault-tolerant error correction with arbitrary distance codes”, Quantum 2, 53 (2018) arXiv:1708.02246 DOI
[17]: N. Delfosse, “Decoding color codes by projection onto surface codes”, Physical Review A 89, (2014) arXiv:1308.6207 DOI
[18]: 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
[19]: 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
[20]: H. Nishimori, “Geometry-Induced Phase Transition in the ±JIsing Model”, Journal of the Physical Society of Japan 55, 3305 (1986) DOI
[21]: 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
[22]: M. Ohzeki, “Accuracy thresholds of topological color codes on the hexagonal and square-octagonal lattices”, Physical Review E 80, (2009) arXiv:0903.2102 DOI
[23]: 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
[24]: D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
[25]: M. Lin and K. Noh, “Exploring the quantum capacity of a Gaussian random displacement channel using Gottesman-Kitaev-Preskill codes and maximum likelihood decoding”, (2024) arXiv:2411.04277
[26]: T. Farrelly, D. K. Tuckett, and T. M. Stace, “Local tensor-network codes”, New Journal of Physics 24, 043015 (2022) arXiv:2109.11996 DOI
[27]: M. Davydova, N. Tantivasadakarn, S. Balasubramanian, and D. Aasen, “Quantum computation from dynamic automorphism codes”, Quantum 8, 1448 (2024) arXiv:2307.10353 DOI
[28]: 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
[29]: 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

Page edit log

Eric Huang (2024-03-18) — most recent
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.), 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.