\([[6,4,2]]\) error-detecting code[14] 

Description

Error-detecting six-qubit code with rate \(2/3\) whose codewords are cat/GHz states. A set of stabilizer generators is \(XXXXXX\) and \(ZZZZZZ\). It is the unique code for its parameters, up to local equivalence [5; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [6].

Stabilizer generators are shown in Figure I. See [6; Appx. B] for a set of logical Paulis.

Figure I: Stabilizer generators of the \([[6,4,2]]\) error-detecting code.

Encoding

Transversal Gates

CNOT and Hadamard gates [6; Appx. B].A \(CZ\) gate implemented by transversal \(S\) and \(S^{\dagger}\) [7]; see also [8].

Gates

Universal Clifford gates via the logical circuit synthesis algorithm [9][10; Sec. III].

Decoding

Efficient decoder for the many-hypercube code [6].

Realizations

Trapped-ion devices: Bayesian quantum phase estimation on a device by Quantinuum [11].

Parents

Cousin

  • Concatenated quantum code — Concatenations of this code with itself yield the level-\(r\) \([[6^r,4^r,2^r]]\) many-hypercube code [6]. The \([[6,4,2]]\) code can be concatenated with the surface code to yield the 6.6.6 color code [12; Appx. A].

References

[1]
A. M. Steane, “Simple quantum error-correcting codes”, Physical Review A 54, 4741 (1996) arXiv:quant-ph/9605021 DOI
[2]
D. Gottesman, “Theory of fault-tolerant quantum computation”, Physical Review A 57, 127 (1998) arXiv:quant-ph/9702029 DOI
[3]
E. Knill, “Fault-Tolerant Postselected Quantum Computation: Schemes”, (2004) arXiv:quant-ph/0402171
[4]
E. Knill, “Fault-Tolerant Postselected Quantum Computation: Threshold Analysis”, (2004) arXiv:quant-ph/0404104
[5]
A. R. Calderbank et al., “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
[6]
H. Goto, “Many-hypercube codes: High-rate quantum error-correcting codes for high-performance fault-tolerant quantum computation”, (2024) arXiv:2403.16054
[7]
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
[8]
M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
[9]
N. Rengaswamy et al., “Logical Clifford Synthesis for Stabilizer Codes”, IEEE Transactions on Quantum Engineering 1, 1 (2020) arXiv:1907.00310 DOI
[10]
N. Rengaswamy et al., “Synthesis of Logical Clifford Operators via Symplectic Geometry”, 2018 IEEE International Symposium on Information Theory (ISIT) (2018) arXiv:1803.06987 DOI
[11]
K. Yamamoto et al., “Demonstrating Bayesian quantum phase estimation with quantum error detection”, Physical Review Research 6, (2024) arXiv:2306.16608 DOI
[12]
B. Criger and B. Terhal, “Noise thresholds for the [4,2,2]-concatenated toric code”, Quantum Information and Computation 16, 1261 (2016) arXiv:1604.04062 DOI
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Zoo Code ID: stab_6_4_2

Cite as:
\([[6,4,2]]\) error-detecting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/stab_6_4_2
BibTeX:
@incollection{eczoo_stab_6_4_2, title={\([[6,4,2]]\) error-detecting code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/stab_6_4_2} }
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\([[6,4,2]]\) error-detecting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/stab_6_4_2

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/stab_6_4_2/stab_6_4_2.yml.