\([[5,1,3]]_q\) Galois-qudit code

\([[5,1,3]]_q\) Galois-qudit code[1]

Description

True stabilizer code that generalizes the five-qubit perfect code to Galois qudits of prime-power dimension \(q=p^m\). It has \(4(m-1)\) stabilizer generators expressed as \(X_{\gamma} Z_{\gamma} Z_{-\gamma} X_{-\gamma} I\) and its cyclic permutations, with \(\gamma\) iterating over basis elements of \(GF(q)\) over \(GF(p)\).

Notes

This code is described in a talk by Gottesman.

Cousin

\([[10,1,4]]_{G}\) tenfold code— The \([[10,1,4]]_{G}\) Abelian group code for \(G=GF(q)\) is defined using a graph that is closely related to the \([[5,1,3]]_{q}\) Galois-qudit code [2].

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

Parents

True Galois-qudit stabilizer code

Graph quantum codeStabilizer Hamiltonian-based QECC Quantum

The \([[5,1,3]]_q\) code is equivalent via a single-Galois-qudit Clifford circuit to a graph quantum code for the group \(G=GF(q)\) [2].

Cyclic quantum codeQECC Quantum

Small-distance block quantum codeQECC Quantum

\([[5,1,3]]_q\) Galois-qudit code

Children

Five-qubit perfect code

The \([[5,1,3]]_q\) Galois-qudit code for \(q=2\) reduces to the five-qubit perfect code.

References

[1]: D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
[2]: D. Schlingemann and R. F. Werner, “Quantum error-correcting codes associated with graphs”, Physical Review A 65, (2001) arXiv:quant-ph/0012111 DOI

Page edit log

Sarah Meng Li (2022-02-21) — most recent
Victor V. Albert (2022-02-21)
Victor V. Albert (2023-01-14)

Cite as:

“\([[5,1,3]]_q\) Galois-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_5_1_3

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits_galois/small/galois_5_1_3.yml.