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
Page edit log
- Victor V. Albert (2023-01-24) — most recent
Cite as:
“Doubled color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/doubled_color