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].Protection
For qubit quantum QR codes obtained from extended binary QR codes, explicit examples satisfy \(n \leq d^2-d+1\) and \(d \leq 4\lfloor (n+1)/24 \rfloor + 3\) [4].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.Cousins
- Quadratic-residue (QR) code— Quantum quadratic-residue codes are quantum analogues of \(q\)-ary quadratic-residue codes.
- Quantum maximum-distance-separable (MDS) code— Almost all quantum QR codes for prime-dimensional qudits are quantum MDS [1; Corr. 11].
- Triorthogonal code— Qubit quantum QR codes are doubly even and admit transversal implementations of the single-qubit Clifford group [4]. They yield a family of high-distance triorthogonal and weak triply even codes via the doubling transformation [4]; such codes admit transversal implementations of the \(T\) gate.
- Quantum divisible code— Qubit quantum QR codes are doubly even and admit transversal implementations of the single-qubit Clifford group [4]. They yield a family of high-distance triorthogonal and weak triply even codes via the doubling transformation [4]; such codes admit transversal implementations of the \(T\) gate.
- Quantum data-syndrome (QDS) code— CSS QDS codes can be constructed from dual-containing cyclic codes without reducing distance; for \(p=8j-1\), quantum QR codes yield \([[p,1,d:r]]\) QDS codes with \(r\leq p+1\) [6; Thms. 13,14].
Primary Hierarchy
Parents
Quantum QR codes are quantum duadic codes since QR codes are duadic codes.
Quantum quadratic-residue (QR) code
Children
The qutrit Golay code is a qutrit quantum QR code since the ternary Golay code is a QR code.
References
- [1]
- E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048
- [2]
- A. Ketkar, A. Klappenecker, S. Kumar, and P. K. Sarvepalli, “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, “Transversal Clifford and T-Gate Codes of Short Length and High Distance”, IEEE Journal on Selected Areas in Information Theory 6, 127 (2025) arXiv:2408.12752 DOI
- [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
- [7]
- A. W. Cross, D. P. DiVincenzo, and B. M. Terhal, “A comparative code study for quantum fault-tolerance”, (2009) arXiv:0711.1556
Page edit log
- Victor V. Albert (2024-05-05) — most recent
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