Doubled color code[13] 

Description

Family of \([[2t^3+8t^2+6t-1,1,2t+1]]\) subsystem color codes (with \(t\geq 1\)), constructed using a generalization of the doubling transformation [4], that admit a Clifford + \(T\) transversal gate set using gauge fixing.

Transversal Gates

Doubled color codes are triply even, so they yield a transversal \(T\) gate [1]. Using gauge fixing, the codes admit a Clifford + \(T\) transversal gate set.

Decoding

ML decoder that can utilize a history of syndromes, based on the Walsh-Hadamard transform [1].

Parents

Cousins

  • Quantum divisible code — Doubled color codes are subsystem codes constructed using a generalization of the doubling transformation [4] that combines doubly even linear binary codes to make triply even codes. The doubling transformation is a special case of level lifting (from two to three) [5; Sec. VI.D].
  • Square-octagon (4.8.8) color code — Doubled color codes can be obtained by pipelining [6] 4.8.8 color codes [5; Sec. VI.D].
  • \([[49,1,5]]\) triorthogonal code — The \([[49,1,5]]\) triorthogonal code can be viewed as a (gauge-fixed) doubled color code obtained from the \([[17,1,5]]\) 2D color code via the doubling transformation [1].
  • \([[15,1,3]]\) quantum Reed-Muller code — The \([[15,1,3]]\) code can be viewed as a (gauge-fixed) doubled color code obtained from the Steane code via the doubling transformation [1].

References

[1]
S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
[2]
T. Jochym-O’Connor and S. D. Bartlett, “Stacked codes: Universal fault-tolerant quantum computation in a two-dimensional layout”, Physical Review A 93, (2016) arXiv:1509.04255 DOI
[3]
C. Jones, P. Brooks, and J. Harrington, “Gauge color codes in two dimensions”, Physical Review A 93, (2016) arXiv:1512.04193 DOI
[4]
K. Betsumiya and A. Munemasa, “On triply even binary codes”, Journal of the London Mathematical Society 86, 1 (2012) arXiv:1012.4134 DOI
[5]
J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI
[6]
J. Haah et al., “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017) arXiv:1703.07847 DOI
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Zoo Code ID: doubled_color

Cite as:
“Doubled color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/doubled_color
BibTeX:
@incollection{eczoo_doubled_color, title={Doubled color code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/doubled_color} }
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Cite as:

“Doubled color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/doubled_color

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/subsystem/topological/color/doubled_color.yml.