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.
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].
Fault Tolerance
Code Capacity Threshold
Independent \(X,Z\) noise: \(p_X = 7.8\%\) under message-passing decoder [8], \(8.7\%\) under projection decoder [16], \(\geq 6\%\) under rescaling decoder [17], \(9.0\%\) under Möbius matching decoder [10], \(10.1\%\) under MaxSAT-based decoder [14], and \(8.2\%\) under concatenated MWPM 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 [19,20], calculated to be \(10.9(2)\%\) in Ref. [20] and \(10.97(1)\%\) in Ref. [21].Depolarizing channel: \(12.6\%\) under the restriction decoder [3] and the projection decoder [16], 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 [18].A measurement threshold of one [22].
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 [23; 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.
- Tensor-network code — Larger triangular color codes can be constructed by contracting legs of tensors of smaller codes [24; 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 [25].
- 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.
- 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
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- H. Bombin and M. A. Martin-Delgado, “Topological Quantum Distillation”, Physical Review Letters 97, (2006) arXiv:quant-ph/0605138 DOI
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- A. J. Landahl, J. T. Anderson, and P. R. Rice, “Fault-tolerant quantum computing with color codes”, (2011) arXiv:1108.5738
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- C. Chamberland et al., “Triangular color codes on trivalent graphs with flag qubits”, New Journal of Physics 22, 023019 (2020) arXiv:1911.00355 DOI
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- F. Thomsen et al., “Low-overhead quantum computing with the color code”, (2024) arXiv:2201.07806
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- 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
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- S.-H. Lee et al., “Low-overhead magic state distillation with color codes”, (2024) arXiv:2409.07707
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- S.-H. Lee, A. Li, and S. D. Bartlett, “Color code decoder with improved scaling for correcting circuit-level noise”, (2024) arXiv:2404.07482
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- H. Nishimori, “Geometry-Induced Phase Transition in the ±JIsing Model”, Journal of the Physical Society of Japan 55, 3305 (1986) DOI
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- 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
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- M. Ohzeki, “Accuracy thresholds of topological color codes on the hexagonal and square-octagonal lattices”, Physical Review E 80, (2009) arXiv:0903.2102 DOI
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- D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
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- 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
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- T. Farrelly, D. K. Tuckett, and T. M. Stace, “Local tensor-network codes”, New Journal of Physics 24, 043015 (2022) arXiv:2109.11996 DOI
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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