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.
Different boundaries affect the logical dimension [3].
Protection
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 [1,4].Single-qubit Clifford and CNOT gates between qubits encoded in holes in the lattice can be implemented via braiding [5].
Gates
Color-code lattice surgery [6].Lattice surgery scheme for a hybrid 6.6.6-4.8.8 layout yields lower resource overhead when compared to analogous surface code scheme [7].
Decoding
Fault-tolerant syndrome extraction circuits [4].Matching decoder [6,8].Integer-program (IP) decoder [4].Two-copy surface-code decoder [9].
Fault Tolerance
Code Capacity Threshold
Independent \(X,Z\) noise: \(p_X = 10.56(1)\%\) under IP decoder [4], \(8.87\%\) under matching decoder [8], \(7.60(2)\%\) under projection decoder [10], and \(8.7\%\) under two-copy surface-code decoder [9] (see [4; 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 [11,12], calculated to be \(10.9(2)\%\) in Ref. [12] and \(10.925(5)\%\) in Ref. [13].
Threshold
Phenomenological noise: \(3.05(4)\%\) under IP decoder [4; Table I] and \(2.08(1)\%\) under projection decoder [10].Circuit-level noise: \(0.082(3)\%\) under IP decoder, \(0.143(1)\%\) under projection decoder [10], \(0.143\%\) under matching decoder [6], and an analytic lower bound of \(\approx 0.1\%\) [8] (see [4; Table I]).
Parent
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 [7].
- \(\mathbb{Z}^n\) hypercubic lattice — 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 [14].
- \([[49,1,5]]\) 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 [15].
- Union-Jack color code — Adding a vertex to the center of every tile of the 4.8.8 tiling yields a rotated version of the Union-Jack lattice.
- Doubled color code — Doubled color codes can be obtained by pipelining [16] 4.8.8 color codes [17; 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]
- E. B. da Silva and W. S. Soares Junior, “Construction of color codes from polygons”, Journal of Physics Communications 2, 095011 (2018) DOI
- [4]
- A. J. Landahl, J. T. Anderson, and P. R. Rice, “Fault-tolerant quantum computing with color codes”, (2011) arXiv:1108.5738
- [5]
- A. G. Fowler, “Two-dimensional color-code quantum computation”, Physical Review A 83, (2011) arXiv:0806.4827 DOI
- [6]
- A. J. Landahl and C. Ryan-Anderson, “Quantum computing by color-code lattice surgery”, (2014) arXiv:1407.5103
- [7]
- F. Thomsen et al., “Low-overhead quantum computing with the color code”, (2024) arXiv:2201.07806
- [8]
- D. S. Wang et al., “Graphical algorithms and threshold error rates for the 2d colour code”, (2009) arXiv:0907.1708
- [9]
- 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).
- [10]
- A. M. Stephens, “Efficient fault-tolerant decoding of topological color codes”, (2014) arXiv:1402.3037
- [11]
- H. Nishimori, “Geometry-Induced Phase Transition in the ±JIsing Model”, Journal of the Physical Society of Japan 55, 3305 (1986) DOI
- [12]
- 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
- [13]
- M. Ohzeki, “Accuracy thresholds of topological color codes on the hexagonal and square-octagonal lattices”, Physical Review E 80, (2009) arXiv:0903.2102 DOI
- [14]
- J. Zhang et al., “Quantum error correction with the color-Gottesman-Kitaev-Preskill code”, Physical Review A 104, (2021) arXiv:2112.14447 DOI
- [15]
- S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
- [16]
- J. Haah et al., “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017) arXiv:1703.07847 DOI
- [17]
- 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