Square-octagon (4.8.8) color code[1] 


Triangular color code defined on a patch of the 4.8.8 (square-octagon) tiling, which itself is obtained by applying a fattening procedure to the square lattice [2].

Stabilizer generators are shown in Figure I.

Figure I: Stabilizer generators of the 4.8.8 color code.


There is a \([[(d^2-1)/2+d, 1, d]]\) code family for any odd distance \(d\) [3; Fig. 2].

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 [1,3].Single-qubit Clifford and CNOT gates between qubits encoded in holes in the lattice can be implemented via braiding [4].


Color-code lattice surgery [5].Lattice surgery scheme for a hybrid 6.6.6-4.8.8 layout yields lower resource overhead when compared to analogous surface code scheme [6].


Fault-tolerant syndrome extraction circuits [3].Matching decoder [5,7].Integer-program (IP) decoder [3].Two-copy surface-code decoder [8].

Fault Tolerance

Color-code lattice surgery [5].Fault-tolerant syndrome extraction circuits [3].

Code Capacity Threshold

Independent \(X,Z\) noise: \(p_X = 10.56(1)\%\) under IP decoder [3], \(8.87\%\) under matching decoder [7], \(7.60(2)\%\) under projection decoder [9], and \(8.7\%\) under two-copy surface-code decoder [8] (see [3; Table I]). The threshold under ML decoding corresponds to the value of critical point of the two-dimensional random-bond three-body Ising model on the Nishimori line [10,11], calculated to be \(10.9(2)\%\) in Ref. [11] and \(10.925(5)\%\) in Ref. [12].


Phenomenological noise: \(3.05(4)\%\) under IP decoder [3; Table I] and \(2.08(1)\%\) under projection decoder [9].Circuit-level noise: \(0.082(3)\%\) under IP decoder, \(0.143(1)\%\) under projection decoder [9], \(0.143\%\) under matching decoder [5], and an analytic lower bound of \(\sim 0.1\%\) [7] (see [3; Table I]).




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Zoo Code ID: 488_color

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“Square-octagon (4.8.8) color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/488_color
@incollection{eczoo_488_color, title={Square-octagon (4.8.8) color code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/488_color} }
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“Square-octagon (4.8.8) color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/488_color

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/color/2d_color/488_color/488_color.yml.