## Description

Stabilizer code designed to correct both data qubit errors and syndrome measurement errors simultaneously due to extra redundancy in its stabilizer generators.

The redundancy can be added to any \([[n,n-m]]\) qubit stabilizer code by expanding its stabilizer generator matrix \(H\) as \begin{align} H_{DS}=\begin{pmatrix}H & I_{m} & 0\\ 0 & A^{T} & I_{r} \end{pmatrix}~, \tag*{(1)}\end{align} where the redundancy is provided by the underlying \([m+r,m]\) syndrome measurement code with generator matrix \(G= (I_m|A)\) [5].

## Protection

Protects against both physical qubit and syndrome measurement errors. Quantum Singleton bounds, quantum Hamming bounds, and quantum MacWilliams identities can be extended to QDS codes. Single-error-correcting QDS codes stemming from impure stabilizer codes must satisfy a variant of the quantum Hamming bound [6].

## Decoding

Syndrome errors are decoded using redundant stabilizer measurements.

## Fault Tolerance

Shor error correction [7,8], in which fault tolerance against syndrome extraction errors is ensured by simply repeating syndrome measurements \(\ell\) times, can be recast as a QDS code whose underlying matrix \(A\) is the identity matrix \(I_m\) repeated \(\ell\) times [5].

## Parent

- Qubit stabilizer code — QDS codes are stabilizer codes whose stabilizer generators encode extra redundancy (via a linear binary code) so as to protect from syndrome measurement errors.

## Children

- \([[7,1,3]]\) Steane code — There exists a set of stabilizer generators for the Steane code that make it a QDS code [1].
- Quantum Golay code — There exists a set of stabilizer generators for the qubit Golay code that make it a QDS code [5].

## Cousins

- \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code — Codes such as the quantum Hamming code can be expanded to QDS codes using almost any good binary linear code because their stabilizer generators all have the same weight [5].
- Quantum maximum-distance-separable (MDS) code — The quantum Singleton bound can be extended to QDS codes [5].
- Perfect quantum code — The quantum Hamming bound can be extended to QDS codes [5].
- Linear binary code — The QDS code construction employs a particular binary linear code to provide protection against syndrome measurement errors.
- Quantum convolutional code — The QDS code framework has been extended to quantum convolutional codes [9].
- Single-shot code — QDS codes contain redundancy in their stabilizer generators so as to protect from syndrome measurement errors.
- Binary BCH code — Primitive narrow-sense BCH codes can be used as the syndrome measurement codes of a QDS code [10]. This construction requires fewer measurements than a previous general construction [1].
- Subsystem qubit stabilizer code — The DS construction can be extended to qubit subsystem qubit stabilizer codes [6].
- Quantum quadratic-residue (QR) code — A family of qubit quantum QR codes can be made into QDS codes [5; Thm. 14].

## References

- [1]
- Y. Fujiwara, “Ability of stabilizer quantum error correction to protect itself from its own imperfection”, Physical Review A 90, (2014) arXiv:1409.2559 DOI
- [2]
- A. Ashikhmin, C.-Y. Lai, and T. A. Brun, “Robust quantum error syndrome extraction by classical coding”, 2014 IEEE International Symposium on Information Theory (2014) DOI
- [3]
- Y. Fujiwara, “Global stabilizer quantum error correction with combinatorial arrays”, 2015 IEEE International Symposium on Information Theory (ISIT) (2015) DOI
- [4]
- A. Ashikhmin, C.-Y. Lai, and T. A. Brun, “Correction of data and syndrome errors by stabilizer codes”, 2016 IEEE International Symposium on Information Theory (ISIT) (2016) DOI
- [5]
- A. Ashikhmin, C.-Y. Lai, and T. A. Brun, “Quantum Data-Syndrome Codes”, IEEE Journal on Selected Areas in Communications 38, 449 (2020) arXiv:1907.01393 DOI
- [6]
- A. Nemec, “Quantum Data-Syndrome Codes: Subsystem and Impure Code Constructions”, (2023) arXiv:2302.01527
- [7]
- P. W. Shor, “Fault-tolerant quantum computation”, (1997) arXiv:quant-ph/9605011
- [8]
- D. P. DiVincenzo and P. W. Shor, “Fault-Tolerant Error Correction with Efficient Quantum Codes”, Physical Review Letters 77, 3260 (1996) arXiv:quant-ph/9605031 DOI
- [9]
- W. Zeng et al., “Quantum convolutional data-syndrome codes”, 2019 IEEE 20th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC) (2019) arXiv:1902.07395 DOI
- [10]
- E. Guttentag, A. Nemec, and K. R. Brown, “Robust Syndrome Extraction via BCH Encoding”, (2023) arXiv:2311.16044

## Page edit log

- Victor V. Albert (2024-05-05) — most recent

## Cite as:

“Quantum data-syndrome (QDS) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/data_syndrome