True Galois-qudit stabilizer code[1–3] 

Description

Also called a linear stabilizer code. A \([[n,k,d]]_q\) stabilizer code whose stabilizer's symplectic representation forms a linear subspace. In other words, the set of \(q\)-ary vectors representing the stabilizer group is closed under both addition and multiplication by elements of \(GF(q)\). In contrast, Galois-qudit stabilizer codes admit sets of vectors that are closed under addition only.

The number of generators \(r\) for a true stabilizer code is a multiple of \(m\) (recall that \(q=p^m\) for Galois qudits). As a result, the number \(k=n-r/m\) of logical qudits is an integer.

Each code can be represented by a stabilizer generator matrix \(H=(A|B)\), where each row \((a|b)\) is the \(GF(q)\)-valued symplectic representation of a stabilizer generator.

A Hermitian self-orthogonal linear \([n,k,d]_{q^2}\) code can be used to construct an \([[n,n-2k]]_q\) true stabilizer code with distance no less than \(d\). This Hermitian construction was first proven via the symplectic representation (showing self-orthogonality under the trace-symplectic inner product; see Ref. [1], Corr. 1), and later proven via the stabilizer-over-\(GF(q^2)\) construction (showing self-orthogonality under the trace-alternating inner product; see Ref. [2], Corr. 19). There is an isomorphism between the symplectic and stabilizer-over-\(GF(q^2)\) representations [4; Thm. 27.3.8]. The Hermitian construction has been extended to \(q^{2m}\)-ary Hermitian self-orthogonal linear codes [5] and similar constructions exist [6].

Protection

Detects errors on up to \(d-1\) qudits, and corrects erasure errors on up to \(d-1\) qudits.

Notes

See Ref. [4,7] for introductions to various stabilizer code constructions.

Parent

Children

Cousins

  • Linear \(q\)-ary code — A true Galois-qudit stabilizer code is the closest quantum analogue of a linear code over \(GF(q)\) because the \(q\)-ary vectors corresponding to the symplectic representation of the stabilizers form a linear subspace.
  • Dual linear code — Hermitian self-orthogonal linear codes over \(GF(q^2)\) yield true stabilizer codes via either the symplectic representation (showing self-orthogonality under the trace-symplectic inner product; see Ref. [1], Corr. 1) or the stabilizer-over-\(GF(q^2)\) construction (showing self-orthogonality under the trace-alternating inner product; see [2; Corr. 19][4; Thm. 27.3.8].
  • Quantum maximum-distance-separable (MDS) code — Many MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction [8–11], in particular from cyclic [12], constacyclic [13,14] and negacyclic [15] codes.
  • Matrix-product code — Hermitian self-orthogonal matrix-product codes over \(GF(q^2)\) can be used to construct true stabilizer codes [16,17].

References

[1]
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
[2]
A. Ketkar et al., “Nonbinary stabilizer codes over finite fields”, (2005) arXiv:quant-ph/0508070
[3]
D. Gottesman. Surviving as a quantum computer in a classical world
[4]
M. F. Ezerman, "Quantum Error-Control Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[5]
C. Galindo and F. Hernando, “On the generalization of the construction of quantum codes from Hermitian self-orthogonal codes”, (2021) arXiv:2012.11998
[6]
M. F. Ezerman, S. Ling, and P. Sole, “Additive Asymmetric Quantum Codes”, IEEE Transactions on Information Theory 57, 5536 (2011) arXiv:1002.4088 DOI
[7]
A. Klappenecker, “Algebraic quantum coding theory”, Quantum Error Correction 307 (2013) DOI
[8]
M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 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]
G. G. La Guardia, “New Quantum MDS Codes”, IEEE Transactions on Information Theory 57, 5551 (2011) DOI
[13]
X. Kai, S. Zhu, and P. Li, “Constacyclic Codes and Some New Quantum MDS Codes”, IEEE Transactions on Information Theory 60, 2080 (2014) DOI
[14]
B. Chen, S. Ling, and G. Zhang, “Application of Constacyclic Codes to Quantum MDS Codes”, IEEE Transactions on Information Theory 61, 1474 (2015) DOI
[15]
X. Kai and S. Zhu, “New Quantum MDS Codes From Negacyclic Codes”, IEEE Transactions on Information Theory 59, 1193 (2013) DOI
[16]
M. Cao and J. Cui, “Construction of new quantum codes via Hermitian dual-containing matrix-product codes”, Quantum Information Processing 19, (2020) DOI
[17]
X. Liu, H. Liu, and L. Yu, “On New Quantum Codes From Matrix Product Codes”, (2021) arXiv:1604.05823
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Zoo Code ID: galois_true_stabilizer

Cite as:
“True Galois-qudit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_true_stabilizer
BibTeX:
@incollection{eczoo_galois_true_stabilizer,
  title={True Galois-qudit stabilizer code},
  booktitle={The Error Correction Zoo},
  year={2022},
  editor={Albert, Victor V. and Faist, Philippe},
  url={https://errorcorrectionzoo.org/c/galois_true_stabilizer}
}
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“True Galois-qudit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_true_stabilizer

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qudits_galois/stabilizer/galois_true_stabilizer.yml.