\([[8,3,2]]\) code[1,2] 

Description

Smallest 3D color code whose physical qubits lie on vertices of a cube and which admits a transversal CCZ gate. Similar constructions exist on \(d\)-dimensional hypercubes and are called hyperoctahedron \([[2^d,d,2]]\) codes [3].

Transversal Gates

CZ gates between any two logical qubits [4] and CCZ gate [1,2,4].

Fault Tolerance

CCZ gate can be distilled in a fault-tolerant manner [5].

Parents

Cousins

  • \([[15,1,3]]\) quantum Reed-Muller code — The \([[8,3,2]]\) code can be obtained from a subset of physical qubits of the \([[15,1,3]]\) code [3].
  • \([[4,2,2]]\) CSS code — The \([[4,2,2]]\) (\([[8,3,2]]\)) code's physical qubits correspond to vertices of a square (cube). Similar constructions exist on \(d\)-dimensional hypercubes and are called hyperoctahedron \([[2^d,d,2]]\) codes [3].

References

[1]
A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
[2]
E. Campbell, “The smallest interesting colour code,” Online available at https://earltcampbell.com/2016/09/26/the-smallest-interesting-colour-code/ (2016), accessed on 2019-12-09.
[3]
M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
[4]
H. Chen et al., “Automated discovery of logical gates for quantum error correction (with Supplementary (153 pages))”, Quantum Information and Computation 22, 947 (2022) arXiv:1912.10063 DOI
[5]
J. Haah and M. B. Hastings, “Measurement sequences for magic state distillation”, Quantum 5, 383 (2021) arXiv:2007.07929 DOI
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Zoo Code ID: stab_8_3_2

Cite as:
“\([[8,3,2]]\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stab_8_3_2
BibTeX:
@incollection{eczoo_stab_8_3_2,
  title={\([[8,3,2]]\) code},
  booktitle={The Error Correction Zoo},
  year={2022},
  editor={Albert, Victor V. and Faist, Philippe},
  url={https://errorcorrectionzoo.org/c/stab_8_3_2}
}
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Permanent link:
https://errorcorrectionzoo.org/c/stab_8_3_2

Cite as:

“\([[8,3,2]]\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stab_8_3_2

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/small_distance/small/stab_8_3_2.yml.