Honeycomb (6.6.6) color code

Honeycomb (6.6.6) color code[1]

Description

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.

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

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 [23].

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.

Parent

2D 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 tiling. The \([[6,4,2]]\) code can be concatenated with the surface code to yield the 6.6.6 color code [24; 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.

Cousins

\(A_2\) hexagonal lattice — 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].
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 [27].
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.
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.

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”, (2024) arXiv:2201.07806
[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”, (2024) 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) (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”, (2024) arXiv:2404.07482
[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]: D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
[24]: 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
[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

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.