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. An \([[n,k,d:r]]\) QDS code corrects any combination of \(t_{\mathrm{D}}\) data-qubit errors and \(t_{\mathrm{S}}\) syndrome-bit errors whenever \(t_{\mathrm{D}}+t_{\mathrm{S}}<d/2\), and its distance cannot exceed that of the underlying stabilizer code [5].
Random QDS codes with \(r\leq n-k\) can attain the stabilizer Gilbert-Varshamov bound [5; Thm. 12]. 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].
Gates
Fault-tolerant flag-based non-transversal logical gates [7].Decoding
Syndrome errors are decoded using redundant stabilizer measurements.Syndrome-measurement codes can outperform repeated syndrome extraction; for the Steane code, a \([15,3]\) syndrome-measurement code uses the same 15 measurements as five-fold repetition of three syndrome bits while achieving lower syndrome-decoding error [5; Fig. 1].Fault Tolerance
Shor error correction [8,9], 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].Fault-tolerant flag-based non-transversal logical gates [7].Notes
QDS codes can be used to estimate physical Pauli noise up to their pure distance [10], and logical Pauli noise for any correctable physical noise [11].Cousins
- \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code— Because every stabilizer generator has the same weight \(2^{r-1}\), quantum Hamming codes admit QDS extensions based on good binary syndrome-measurement codes [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 [12].
- Single-shot code— QDS codes are closely related to single-shot codes because both use redundant syndrome information to suppress measurement errors in a single round of syndrome extraction [13].
- Primitive narrow-sense BCH code— Primitive narrow-sense BCH codes can be used as the syndrome measurement codes of a QDS code [14]. This construction requires fewer measurements than a previous general construction [1].
- Subsystem qubit stabilizer code— The DS construction can be extended to subsystem qubit stabilizer codes [6].
- Quantum quadratic-residue (QR) code— CSS QDS codes can be constructed from dual-containing cyclic codes without reducing distance; for \(p=8j-1\), quantum QR codes yield \([[p,1,d:r]]\) QDS codes with \(r\leq p+1\) [5; Thms. 13,14].
Primary Hierarchy
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) 1114 (2015) DOI
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- 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
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- 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
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- A. Nemec, “Quantum Data-Syndrome Codes: Subsystem and Impure Code Constructions”, (2023) arXiv:2302.01527
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- B. Anker and M. Marvian, “Universal Fault Tolerance with Non-Transversal Clifford Gates”, (2025) arXiv:2510.08402
- [8]
- P. W. Shor, “Fault-tolerant quantum computation”, (1997) arXiv:quant-ph/9605011
- [9]
- 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
- [10]
- T. Wagner, H. Kampermann, D. Bruß, and M. Kliesch, “Pauli channels can be estimated from syndrome measurements in quantum error correction”, Quantum 6, 809 (2022) arXiv:2107.14252 DOI
- [11]
- T. Wagner, H. Kampermann, D. Bruß, and M. Kliesch, “Learning Logical Pauli Noise in Quantum Error Correction”, Physical Review Letters 130, (2023) arXiv:2209.09267 DOI
- [12]
- W. Zeng, A. Ashikhmin, M. Woolls, and L. P. Pryadko, “Quantum convolutional data-syndrome codes”, 2019 IEEE 20th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC) (2019) arXiv:1902.07395 DOI
- [13]
- E. T. Campbell, “A theory of single-shot error correction for adversarial noise”, Quantum Science and Technology 4, 025006 (2019) arXiv:1805.09271 DOI
- [14]
- E. Guttentag, A. Nemec, and K. R. Brown, “Robust Syndrome Extraction via BCH Encoding”, 2024 IEEE International Symposium on Information Theory (ISIT) 2281 (2024) arXiv:2311.16044 DOI
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