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 MDS 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.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 [7].
- Hermitian Galois-qudit code— Many quantum MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction [8–11], in particular from cyclic [7], 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 [8–11], in particular from cyclic [7], 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].
- Good QLDPC code— AEL distance amplification [19,20] can be used to construct asymptotically good QLDPC codes that approach the quantum Singleton bound [21; Corr. 5.3].
- Asymmetric quantum code— An asymmetric Singleton bound and linear programming bounds for asymmetric CSS codes have been formulated [22]. Asymmetric MDS codes have been characterized [23].
- Perfect-tensor code— AME states for even \(n\) are examples of quantum MDS codes with no logical qubits [24–26]. A family of conjectured perfect-tensor codes is quantum MDS [8].
- Quantum data-syndrome (QDS) code— The quantum Singleton bound can be extended to QDS codes [27].
- 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 [28].
- 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 [29].
Primary Hierarchy
Parents
Quantum maximum-distance-separable (MDS) code
Children
The \([[2m,2m-2,2]]\) error-detecting code forms one of the two families qubit quantum MDS codes [8].
The five-qubit code is one of the two qubit quantum MDS codes.
The three-qutrit code is the smallest nontrivial quantum MDS code.
References
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- A. Winter, private communication (2019).
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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