True Galois-qudit stabilizer code

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

Galois-qudit stabilizer code

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

Page edit log

Victor V. Albert (2022-07-22) — most recent
Daniel Gottesman (2022-02-17)
Victor V. Albert (2022-02-17)

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.