Square-octagon (4.8.8) color code

Square-octagon (4.8.8) color code[1]

Description

2D 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]. An equivalent description uses the Tetrakis square tiling (a.k.a. the Union Jack lattice), which is dual to the 4.8.8 lattice [3]. Among the three semiregular triangular 2D color-code families, the 4.8.8 family uses the fewest physical qubits for a given distance and is the only one of the three with transversal implementations of the full Clifford group [4].

Stabilizer generators are shown in Fig. I.

Figure I: Stabilizer generators of the 4.8.8 color code.

Different boundaries affect the logical dimension [5].

Protection

There is a \([[(d^2-1)/2+d, 1, d]]\) code family for any odd distance \(d\) [4; 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 [1,4].Transversal logical Clifford gates in the Union Jack formulation [3].Single-qubit Clifford and CNOT gates between qubits encoded in holes in the lattice can be implemented via braiding [6].

Gates

Color-code lattice surgery [7].Lattice surgery scheme for a hybrid 6.6.6-4.8.8 layout yields lower resource overhead when compared to analogous surface code scheme [8].

Decoding

Fault-tolerant syndrome extraction circuits [4].Matching decoder [7,9].Integer-program (IP) decoder [4].Two-copy surface-code decoder [10].

Fault Tolerance

Color-code lattice surgery [7].Fault-tolerant syndrome extraction circuits [4].

Code Capacity Threshold

Independent \(X,Z\) noise: \(p_X = 10.56(1)\%\) under IP decoder [4], \(8.87\%\) under matching decoder [9], \(7.60(2)\%\) under projection decoder [11], and \(8.7\%\) under two-copy surface-code decoder [10] (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 [12,13], calculated to be \(10.9(2)\%\) in Ref. [13] (and in the Union Jack formulation in Ref. [3]) and \(10.925(5)\%\) in Ref. [14].

Threshold

Phenomenological noise: \(3.05(4)\%\) under IP decoder [4; Table I] and \(2.08(1)\%\) under projection decoder [11].Circuit-level noise: \(0.082(3)\%\) under IP decoder, \(0.143(1)\%\) under projection decoder [11], \(0.143\%\) under matching decoder [7], and an analytic lower bound of \(\approx 0.1\%\) [9] (see [4; Table I]).

Realizations

Neutral atom arrays: logical magic-state distillation using distance-three and five 4.8.8 color codes, observing an improvement in logical fidelity on a device by Quera [15].

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 [8].
\(\mathbb{Z}^n\) hypercubic lattice— The 4.8.8 (square-octagon) tiling is obtained by applying a fattening procedure to the square lattice [2].
Concatenated GKP code— GKP codes have been concatenated with 4.8.8 color codes [16].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Qubit QLDPC codeStabilizer Hamiltonian-based QECC Quantum

Color codeCSS Stabilizer Hamiltonian-based Qubit QECC Quantum

2D color codeTwist-defect color QLDPC Qubit Generalized homological-product Lattice stabilizer CSS Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum

Parents

2D color code

Square-octagon (4.8.8) color code

Children

\([[4,2,2]]\) Four-qubit code

The \([[4,2,2]]\) code can be interpreted as a 2D color code on a square of the 4.8.8 tiling [17,18]. Removing \(X\) checks from blue octagons and \(Z\) checks from green octagons of the 4.8.8 color code yields a light 4.8.8 color code that is equivalent to concatenating the surface/toric code with the \([[4,2,2]]\) code [19].

\([[17,1,5]]\) 4.8.8 color code

The \([[17,1,5]]\) color code can be defined on the 4.8.8 tiling [20].

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]: H. G. Katzgraber, H. Bombin, R. S. Andrist, and M. A. Martin-Delgado, “Topological color codes on Union Jack lattices: a stable implementation of the whole Clifford group”, Physical Review A 81, (2010) arXiv:0910.0573 DOI
[4]: A. J. Landahl, J. T. Anderson, and P. R. Rice, “Fault-tolerant quantum computing with color codes”, (2011) arXiv:1108.5738
[5]: E. B. da Silva and W. S. Soares Junior, “Construction of color codes from polygons”, Journal of Physics Communications 2, 095011 (2018) DOI
[6]: A. G. Fowler, “Two-dimensional color-code quantum computation”, Physical Review A 83, (2011) arXiv:0806.4827 DOI
[7]: A. J. Landahl and C. Ryan-Anderson, “Quantum computing by color-code lattice surgery”, (2014) arXiv:1407.5103
[8]: F. Thomsen, M. S. Kesselring, S. D. Bartlett, and B. J. Brown, “Low-overhead quantum computing with the color code”, Physical Review Research 6, (2024) arXiv:2201.07806 DOI
[9]: D. S. Wang, A. G. Fowler, C. D. Hill, and L. C. L. Hollenberg, “Graphical algorithms and threshold error rates for the 2d colour code”, (2009) arXiv:0907.1708
[10]: G. Duclos-Cianci, H. Bombın, and D. Poulin, “Fast decoding algorithm for subspace and subsystem color codes and local equivalence of topological phases.” Personal communication (2011)
[11]: A. M. Stephens, “Efficient fault-tolerant decoding of topological color codes”, (2014) arXiv:1402.3037
[12]: H. Nishimori, “Geometry-Induced Phase Transition in the ±JIsing Model”, Journal of the Physical Society of Japan 55, 3305 (1986) DOI
[13]: 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
[14]: M. Ohzeki, “Accuracy thresholds of topological color codes on the hexagonal and square-octagonal lattices”, Physical Review E 80, (2009) arXiv:0903.2102 DOI
[15]: P. Sales Rodriguez et al., “Experimental demonstration of logical magic state distillation”, Nature 645, 620 (2025) arXiv:2412.15165 DOI
[16]: J. Zhang, J. Zhao, Y.-C. Wu, and G.-P. Guo, “Quantum error correction with the color-Gottesman-Kitaev-Preskill code”, Physical Review A 104, (2021) arXiv:2112.14447 DOI
[17]: M. S. Kesselring, J. C. Magdalena de la Fuente, F. Thomsen, J. Eisert, S. D. Bartlett, and B. J. Brown, “Anyon Condensation and the Color Code”, PRX Quantum 5, (2024) arXiv:2212.00042 DOI
[18]: R. S. Gupta et al., “Encoding a magic state with beyond break-even fidelity”, Nature 625, 259 (2024) arXiv:2305.13581 DOI
[19]: 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
[20]: S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239

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.