\([[6,2,2]]\) \(C_6\) 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 yield parameter 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] (see also Ref. [5]).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 [6,7].
Parents
Cousins
- Concatenated qubit code — Concatenations of \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation schemes [1] (see also Ref. [5]) 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 [6,7].
- \([[4,2,2]]\) Four-qubit code — Concatenations of \([[4,2,2]]\) and \([[6,2,2]]\) \(C_6\) codes yield fault-tolerant quantum computation schemes [1] (see also Ref. [5]) 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 [6,7].
- Concatenated GKP code — Recursively concatenating the \([[6,2,2]]\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel [8].
- Quantum Lego code — The \([[6,4,2]]\) error-detecting code can be constructed out of two \([[4,2,2]]\) codes in the quantum Lego code framework.
- \([[4,2,2]]\) Four-qubit code — The \([[6,4,2]]\) error-detecting code can be constructed out of two \([[4,2,2]]\) codes in the quantum Lego code framework.
- \([[2^r-1, 2^r-2r-1, 3]]\) 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 [6].
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]
- J. Cho, “Fault-tolerant linear optics quantum computation by error-detecting quantum state transfer”, Physical Review A 76, (2007) arXiv:quant-ph/0612073 DOI
- [6]
- H. Yamasaki and M. Koashi, “Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation”, Nature Physics 20, 247 (2024) arXiv:2207.08826 DOI
- [7]
- S. Yoshida, S. Tamiya, and H. Yamasaki, “Concatenate codes, save qubits”, (2024) arXiv:2402.09606
- [8]
- K. Fukui et al., “High-Threshold Fault-Tolerant Quantum Computation with Analog Quantum Error Correction”, Physical Review X 8, (2018) arXiv:1712.00294 DOI
Page edit log
- Victor V. Albert (2024-03-28) — most recent
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