True Galois-qudit stabilizer code[1][2][3]

Description

Also called a linear stabilizer code. A \([[n,k,d]]_{GF(q)}\) 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 \([n,k,d]_{GF(q^2)}\) code can be used to construct an \([[n,n-2k]]_{GF(q)}\) 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 (Ref. [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.

Parent

Children

  • Galois-qudit BCH code — Galois-qudit BCH codes constructed via the CSS construction are Galois-qudit CSS codes, and the rest are true stabilizer codes.
  • Galois-qudit CSS code — Galois-qudit CSS codes are true stabilizer codes [3].
  • Galois-qudit GRS code — Galois-qudit GRS codes constructed via the CSS construction are Galois-qudit CSS codes, and the rest are true stabilizer codes.
  • Stabilizer code over \(GF(q^2)\) — Trace-alternating self-orthogonal linear codes over \(GF(q^2)\) are equivalent to a class of true stabilizer codes [3]. Hermitian self-orthogonal linear codes over \(GF(q^2)\) are automatically trace-alternating self-orthogonal and can be used to construct true stabilizer codes via the stabilizer-over-\(GF(q^2)\) construction ([2], Corr. 19).

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 Ref. [2], Corr. 19 or Ref. [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 [7][8][9][10], in particular from cyclic [11], constacyclic [12][13] and negacyclic [14] codes.
  • Matrix-product code — Hermitian self-orthogonal matrix-product codes over \(GF(q^2)\) can be used to construct true stabilizer codes [15][16].

References

[1]
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001). DOI
[2]
Avanti Ketkar et al., “Nonbinary stabilizer codes over finite fields”. quant-ph/0508070
[3]
D. Gottesman. Surviving as a quantum computer in a classical world
[4]
W. C. Huffman, J.-L. Kim, and P. Solé, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021). DOI
[5]
Carlos Galindo and Fernando Hernando, “On the generalization of the construction of quantum codes from Hermitian self-orthogonal codes”. 2012.11998
[6]
M. F. Ezerman, S. Ling, and P. Sole, “Additive Asymmetric Quantum Codes”, IEEE Transactions on Information Theory 57, 5536 (2011). DOI; 1002.4088
[7]
M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004). DOI; quant-ph/0312164
[8]
R. Li and Z. Xu, “Construction of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">]</mml:mo><mml:mo stretchy="false">]</mml:mo><mml:msub><mml:mrow /><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>quantum codes for odd prime power<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math>”, Physical Review A 82, (2010). DOI; 0906.2509
[9]
Xianmang He, Liqing Xu, and Hao Chen, “New $q$-ary Quantum MDS Codes with Distances Bigger than $\frac{q}{2}$”. 1507.08355
[10]
Liangdong Lu et al., “New Quantum MDS codes constructed from Constacyclic codes”. 1803.07927
[11]
G. G. La Guardia, “New Quantum MDS Codes”, IEEE Transactions on Information Theory 57, 5551 (2011). DOI
[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
[13]
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. Cao and J. Cui, “Construction of new quantum codes via Hermitian dual-containing matrix-product codes”, Quantum Information Processing 19, (2020). DOI
[16]
Xiusheng Liu, Hualu Liu, and Long Yu, “On New Quantum Codes From Matrix Product Codes”. 1604.05823
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Internal code ID: galois_true_stabilizer

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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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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

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