Quantum quadratic-residue (QR) code[13] 

Description

Galois-qudit \([[n,1]]_q\) pure self-dual CSS code constructed from a dual-containing QR code via the Galois-qudit CSS construction. For \(q\) not divisible by \(n\), its distance satisfies \(d^2-d+1 \geq n\) when \(n \equiv 3\) modulo 4 [2; Thm. 40] and \(d \geq \sqrt{n}\) when \(n\equiv 1\) modulo 4 [2; Thm. 41].

Transversal Gates

Qubit quantum QR codes admit transversal implementations of the single-qubit Clifford group [4]. They yield a family of high-distance triorthogonal codes [4] via the doubling transformation [5]; such codes admit transversal implementations of the \(T\) gate.

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References

[1]
E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048
[2]
A. Ketkar et al., “Nonbinary stabilizer codes over finite fields”, (2005) arXiv:quant-ph/0508070
[3]
C.-Y. Lai and C.-C. Lu, “A Construction of Quantum Stabilizer Codes Based on Syndrome Assignment by Classical Parity-Check Matrices”, IEEE Transactions on Information Theory 57, 7163 (2011) arXiv:0712.0103 DOI
[4]
S. P. Jain and V. V. Albert, “High-distance codes with transversal Clifford and T-gates”, (2024) arXiv:2408.12752
[5]
S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
[6]
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
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Zoo Code ID: galois_quad_residue

Cite as:
“Quantum quadratic-residue (QR) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/galois_quad_residue
BibTeX:
@incollection{eczoo_galois_quad_residue, title={Quantum quadratic-residue (QR) code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/galois_quad_residue} }
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Cite as:

“Quantum quadratic-residue (QR) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/galois_quad_residue

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits_galois/stabilizer/duadic/galois_quad_residue.yml.