Description
A type of block quantum code whose parameters satisfy the quantum Singleton bound with equality.
An \(((n,K,d))\) code constructed out of \(q\)-dimensional qudits is a quantum MDS code if parameters \(n\), \(k\), \(d\), and \(q\) are such that the quantum Singleton bound [1–3], \begin{align} K \leq q^{n-2(d-1)} \tag*{(1)}\end{align} becomes an equality for such codes. When \(K = q^k\) for some integer \(k\), the above reduces to \(2(d-1) \leq n-k\). Such codes are pure [3]; see also [4] mentioned in Ref. [5]. The length \(n\) of a quantum MSD code with distance \(d \geq 3\) is bounded by the qudit dimension, \(n \leq q^2 + d - 2\) [5].
Protection
Given \(n\) and \(k\), MDS codes have the highest distance possible of all codes and so have the best possible error correction properties.
Notes
Parents
Children
- \([[2m,2m-2,2]]\) error-detecting code — The \([[2m,2m-2,2]]\) error-detecting code forms one of the two families qubit quantum MDS codes [7].
- Five-qubit perfect code — The five-qubit code is one of the two qubit quantum MDS codes.
- Three-qutrit code — The three-qutrit code is the smallest nontrivial quantum MDS code.
- \([[6,2,3]]_{q}\) code
- \([[7,3,3]]_{q}\) code
Cousins
- Maximum distance separable (MDS) code
- Cyclic linear \(q\)-ary code — Quantum MDS codes can be constructed from \(q\)-ary cyclic codes using the Hermitian construction [8].
- Hermitian Galois-qudit code — Many quantum MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction [7,9–11], in particular from cyclic [8], constacyclic [11–13] and negacyclic [14] codes.
- Constacyclic code — Many quantum MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction [7,9–11], in particular from cyclic [8], constacyclic [11–13], and negacyclic [14] codes.
- Generalized RS (GRS) code — Some quantum MDS codes are constructed from cyclic and constacyclic codes [15] which are GRS codes [16,17].
- Skew-cyclic CSS code — Some quantum MDS codes are constructed from cyclic and constacyclic codes using the Galois-qudit CSS construction [18].
- Asymmetric quantum code — An asymmetric Singleton bound and linear programming bounds for asymmetric CSS codes have been formulated [19]. Asymmetric MDS codes have been characterized [20].
- Perfect-tensor code — AME states for even \(n\) are examples of quantum MDS codes with no logical qubits [21,22]. A family of conjectured perfect-tensor codes is quantum MDS [7].
- Good QLDPC code — AEL distance amplification [23,24] can be used to construct asymptotically good QLDPC codes that approach the quantum Singleton bound [25; Corr. 5.3].
- Quantum data-syndrome (QDS) code — The quantum Singleton bound can be extended to QDS codes [26].
- EA MDS code
- Singleton-bound approaching AQECC — Singleton-bound approaching AQECCs saturate the quantum Singleton bound.
- Quantum quadratic-residue (QR) code — Almost all quantum QR codes for prime-dimensional qudits are quantum MDS [3; Corr. 11].
- Galois-qudit quantum RM code — There exists a quantum RM code \([[q, q − 2ν − 2, ν + 2]]_q\) for \( 0\leq v \leq \frac{(q-2)}{2}\) and \([[q^2,q^2-2v-2,v+2]]_q\) for \(0\leq v \leq q-2\). Both these codes satisfy the quantum Singleton bound.
- Galois-qudit GRS code — Some Galois-qudit GRS codes are quantum MDS [27].
- Galois-qudit RS code — A polynomial code is a quantum MDS code when \(n-k_1=k_1-k_2\).
- Subsystem Galois-qudit stabilizer code — All pure MDS subsystem stabilizer codes are derived from MDS stabilizer codes [28].
References
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- E. Knill, R. Laflamme, and L. Viola, “Theory of Quantum Error Correction for General Noise”, Physical Review Letters 84, 2525 (2000) arXiv:quant-ph/9604034 DOI
- [2]
- N. J. Cerf and R. Cleve, “Information-theoretic interpretation of quantum error-correcting codes”, Physical Review A 56, 1721 (1997) arXiv:quant-ph/9702031 DOI
- [3]
- E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048
- [4]
- A. Winter, private communication (2019).
- [5]
- F. Huber and M. Grassl, “Quantum Codes of Maximal Distance and Highly Entangled Subspaces”, Quantum 4, 284 (2020) arXiv:1907.07733 DOI
- [6]
- A. Klappenecker, “Algebraic quantum coding theory”, Quantum Error Correction 307 (2013) DOI
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- M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 DOI
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- G. G. La Guardia, “New Quantum MDS Codes”, IEEE Transactions on Information Theory 57, 5551 (2011) DOI
- [9]
- R. Li and Z. Xu, “Construction of[[n,n−4,3]]qquantum codes for odd prime powerq”, Physical Review A 82, (2010) arXiv:0906.2509 DOI
- [10]
- X. He, L. Xu, and H. Chen, “New \(q\)-ary Quantum MDS Codes with Distances Bigger than \(\frac{q}{2}\)”, (2015) arXiv:1507.08355
- [11]
- L. Lu et al., “New Quantum MDS codes constructed from Constacyclic codes”, (2018) arXiv:1803.07927
- [12]
- X. Kai, S. Zhu, and P. Li, “Constacyclic Codes and Some New Quantum MDS Codes”, IEEE Transactions on Information Theory 60, 2080 (2014) DOI
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- B. Chen, S. Ling, and G. Zhang, “Application of Constacyclic Codes to Quantum MDS Codes”, IEEE Transactions on Information Theory 61, 1474 (2015) DOI
- [14]
- X. Kai and S. Zhu, “New Quantum MDS Codes From Negacyclic Codes”, IEEE Transactions on Information Theory 59, 1193 (2013) DOI
- [15]
- M. Grassl and M. Rotteler, “Quantum MDS codes over small fields”, 2015 IEEE International Symposium on Information Theory (ISIT) (2015) arXiv:1502.05267 DOI
- [16]
- S. Ball, “Grassl–Rötteler cyclic and consta-cyclic MDS codes are generalised Reed–Solomon codes”, Designs, Codes and Cryptography 91, 1685 (2022) DOI
- [17]
- H. Liu and S. Liu, “A class of constacyclic codes are generalized Reed–Solomon codes”, Designs, Codes and Cryptography 91, 4143 (2023) DOI
- [18]
- H. Q. Dinh et al., “A class of skew cyclic codes and application in quantum codes construction”, Discrete Mathematics 344, 112189 (2021) DOI
- [19]
- P. K. Sarvepalli, A. Klappenecker, and M. Rötteler, “Asymmetric quantum codes: constructions, bounds and performance”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, 1645 (2009) DOI
- [20]
- M. F. EZERMAN et al., “PURE ASYMMETRIC QUANTUM MDS CODES FROM CSS CONSTRUCTION: A COMPLETE CHARACTERIZATION”, International Journal of Quantum Information 11, 1350027 (2013) arXiv:1006.1694 DOI
- [21]
- A. J. Scott, “Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions”, Physical Review A 69, (2004) arXiv:quant-ph/0310137 DOI
- [22]
- Z. Raissi et al., “Optimal quantum error correcting codes from absolutely maximally entangled states”, Journal of Physics A: Mathematical and Theoretical 51, 075301 (2018) arXiv:1701.03359 DOI
- [23]
- N. Alon, J. Edmonds, and M. Luby, “Linear time erasure codes with nearly optimal recovery”, Proceedings of IEEE 36th Annual Foundations of Computer Science DOI
- [24]
- N. Alon and M. Luby, “A linear time erasure-resilient code with nearly optimal recovery”, IEEE Transactions on Information Theory 42, 1732 (1996) DOI
- [25]
- T. Bergamaschi, L. Golowich, and S. Gunn, “Approaching the Quantum Singleton Bound with Approximate Error Correction”, (2022) arXiv:2212.09935
- [26]
- 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
- [27]
- L. Jin and C. Xing, “A Construction of New Quantum MDS Codes”, (2020) arXiv:1311.3009
- [28]
- S. A. Aly and A. Klappenecker, “Subsystem Code Constructions”, (2008) arXiv:0712.4321
Page edit log
- Victor V. Albert (2022-07-22) — most recent
- Victor V. Albert (2022-01-10)
- Qingfeng (Kee) Wang (2021-12-20)
Cite as:
“Quantum maximum-distance-separable (MDS) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_mds