Quantum quadratic-residue (QR) code

Quantum quadratic-residue (QR) code[1–3]

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.

Cousins

Quadratic-residue (QR) code
Quantum data-syndrome (QDS) code— A family of qubit quantum QR codes can be made into QDS codes [6; Thm. 14].
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.

Member of code lists

Primary Hierarchy

Quantum Domain

Galois-qudit Kingdom

Galois-qudit codeQECC Quantum

Galois-qudit USt code

Galois-qudit stabilizer codeStabilizer Hamiltonian-based QECC Quantum

True Galois-qudit stabilizer code

Galois-qudit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum

Parents

Galois-qudit CSS code

Quantum duadic codeStabilizer Hamiltonian-based QECC Quantum

Quantum QR codes are quantum duadic codes since QR codes are duadic codes.

Quantum quadratic-residue (QR) code

Children

\([[7,1,3]]\) Steane code

The Steane code is a qubit quantum QR code [3,6].

\([[23, 1, 7]]\) Quantum Golay code

The Golay code is a qubit quantum QR code [3,6].

\([[11,1,5]]_3\) qutrit Golay code

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, “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

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

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