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].

Decoding

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].

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 [16].A measurement threshold of one [20].

Parent

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 [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.

Cousins

  • \(A_2\) 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.

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 et al., “Triangular color codes on trivalent graphs with flag qubits”, New Journal of Physics 22, 023019 (2020) arXiv:1911.00355 DOI
[4]
F. Thomsen et al., “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]
P. Sarvepalli and R. Raussendorf, “Efficient decoding of topological color codes”, Physical Review A 85, (2012) arXiv:1111.0831 DOI
[7]
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
[8]
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
[9]
C. Gidney and C. Jones, “New circuits and an open source decoder for the color code”, (2023) arXiv:2312.08813
[10]
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
[11]
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
[12]
L. Berent et al., “Decoding quantum color codes with MaxSAT”, (2023) arXiv:2303.14237
[13]
C. Chamberland and M. E. Beverland, “Flag fault-tolerant error correction with arbitrary distance codes”, Quantum 2, 53 (2018) arXiv:1708.02246 DOI
[14]
N. Delfosse, “Decoding color codes by projection onto surface codes”, Physical Review A 89, (2014) arXiv:1308.6207 DOI
[15]
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
[16]
S.-H. Lee, A. Li, and S. D. Bartlett, “Color code decoder with improved scaling for correcting circuit-level noise”, (2024) arXiv:2404.07482
[17]
H. Nishimori, “Geometry-Induced Phase Transition in the ±JIsing Model”, Journal of the Physical Society of Japan 55, 3305 (1986) DOI
[18]
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
[19]
M. Ohzeki, “Accuracy thresholds of topological color codes on the hexagonal and square-octagonal lattices”, Physical Review E 80, (2009) arXiv:0903.2102 DOI
[20]
D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
[21]
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
[22]
M. Davydova et al., “Quantum computation from dynamic automorphism codes”, (2023) arXiv:2307.10353
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Zoo Code ID: triangular_color

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
BibTeX:
@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.