Square-octagon (4.8.8) color code

Square-octagon (4.8.8) color code[1]

Description

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.

Protection

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].

Gates

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].

Decoding

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 a critical point of a two-dimensional three-body random-bond Ising model (RBIM) on the Nishimori line [10,11], calculated to be \(10.9(2)\%\) in Ref. [11] and \(10.925(5)\%\) in Ref. [12].

Threshold

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]).

Parent

2D color code

Cousins

Honeycomb (6.6.6) 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 [6].
\(\mathbb{Z}^n\) hypercubic lattice code — The 4.8.8 (square-octagon) tiling is obtained by applying a fattening procedure to the honeycomb lattice [2].
Concatenated GKP code — GKP codes have been concatenated with 4.8.8 color codes [13].
\([[49,1,5]]\) triorthogonal code — The \([[49,1,5]]\) triorthogonal code can be viewed as a (gauge-fixed) doubled color code obtained from the \([[17,1,5]]\) 4.8.8 color code via the doubling transformation [14].
Doubled color code — Doubled color codes can be obtained by pipelining [15] 4.8.8 color codes [16; Sec. VI.D].

References

[1]: H. Bombin and M. A. Martin-Delgado, “Topological Quantum Distillation”, Physical Review Letters 97, (2006) arXiv:quant-ph/0605138 DOI
[2]: H. Bombin and M. A. Martin-Delgado, “Exact topological quantum order inD=3and beyond: Branyons and brane-net condensates”, Physical Review B 75, (2007) arXiv:cond-mat/0607736 DOI
[3]: A. J. Landahl, J. T. Anderson, and P. R. Rice, “Fault-tolerant quantum computing with color codes”, (2011) arXiv:1108.5738
[4]: A. G. Fowler, “Two-dimensional color-code quantum computation”, Physical Review A 83, (2011) arXiv:0806.4827 DOI
[5]: A. J. Landahl and C. Ryan-Anderson, “Quantum computing by color-code lattice surgery”, (2014) arXiv:1407.5103
[6]: F. Thomsen et al., “Low-overhead quantum computing with the color code”, (2024) arXiv:2201.07806
[7]: D. S. Wang et al., “Graphical algorithms and threshold error rates for the 2d colour code”, (2009) arXiv:0907.1708
[8]: Duclos-Cianci, Guillaume, Héctor Bombın, and David Poulin. "Fast decoding algorithm for subspace and subsystem color codes and local equivalence of topological phases." Personal communication (2011).
[9]: A. M. Stephens, “Efficient fault-tolerant decoding of topological color codes”, (2014) arXiv:1402.3037
[10]: H. Nishimori, “Geometry-Induced Phase Transition in the ±JIsing Model”, Journal of the Physical Society of Japan 55, 3305 (1986) DOI
[11]: 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
[12]: M. Ohzeki, “Accuracy thresholds of topological color codes on the hexagonal and square-octagonal lattices”, Physical Review E 80, (2009) arXiv:0903.2102 DOI
[13]: J. Zhang et al., “Quantum error correction with the color-Gottesman-Kitaev-Preskill code”, Physical Review A 104, (2021) arXiv:2112.14447 DOI
[14]: S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
[15]: J. Haah et al., “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017) arXiv:1703.07847 DOI
[16]: J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI

Page edit log

Victor V. Albert (2024-03-29) — most recent
Eric Huang (2024-03-18)

Cite as:

“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.