\([[6,2,2]]\) \(C_6\) code[1] 

Description

Error-detecting self-dual CSS code used in concatenation schemes for fault-tolerant quantum computation. A set of stabilizer generators is \(IIXXXX\) and \(XXIIXX\), together with the same two \(Z\)-type generators.

Magic

Various magic-state distillation protocols exist for the \([[4,2,2]]\) qubit code and the \([[6,2,2]]\) \(C_6\) code in what are known as Meier-Eastin-Knill (MEK) protocols [2]. For example, the magic-state scaling exponent is \(\gamma = \log_2 5 \approx 2.322\) for a protocol using the \([[10,2,2]]\) code [3; Box 2]; see also [4; Table IV].

Fault Tolerance

Concatenations of \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation schemes [1].Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [5,6].

Parents

Cousins

  • Concatenated quantum code — Concatenations of \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation schemes [1] and the Meier-Eastin-Knill (MEK) magic-state distillation protocols [2]. Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [5,6].
  • \([[4,2,2]]\) CSS code — Concatenations of \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation schemes [1] and the Meier-Eastin-Knill (MEK) magic-state distillation protocols [2]. Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [5,6].
  • \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code — Concatenations of quantum Hamming codes with the \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [5].

References

[1]
E. Knill, “Quantum computing with realistically noisy devices”, Nature 434, 39 (2005) arXiv:quant-ph/0410199 DOI
[2]
A. M. Meier, B. Eastin, and E. Knill, “Magic-state distillation with the four-qubit code”, (2012) arXiv:1204.4221
[3]
E. T. Campbell, B. M. Terhal, and C. Vuillot, “Roads towards fault-tolerant universal quantum computation”, Nature 549, 172 (2017) arXiv:1612.07330 DOI
[4]
Quantum Information and Computation 18, (2018) arXiv:1709.02789 DOI
[5]
H. Yamasaki and M. Koashi, “Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation”, Nature Physics 20, 247 (2024) arXiv:2207.08826 DOI
[6]
S. Yoshida, S. Tamiya, and H. Yamasaki, “Concatenate codes, save qubits”, (2024) arXiv:2402.09606
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Zoo Code ID: stab_6_2_2

Cite as:
\([[6,2,2]]\) \(C_6\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/stab_6_2_2
BibTeX:
@incollection{eczoo_stab_6_2_2, title={\([[6,2,2]]\) \(C_6\) code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/stab_6_2_2} }
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Permanent link:
https://errorcorrectionzoo.org/c/stab_6_2_2

Cite as:

\([[6,2,2]]\) \(C_6\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/stab_6_2_2

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/stab_6_2_2.yml.