Perfect quantum code

Description

A non-degenerate code constructed out of \(q\)-dimensional qudits and having parameters \(((n,K,2t+1))\) is perfect if \(n\), \(K\), \(t\), and \(q\) are such that the quantum Hamming bound \begin{align} \sum_{j=0}^{t}(q^2-1)^{j}{n \choose j}\leq q^{n}/K \end{align} becomes an equality. For example, for a qubit \(q=2\) code with one logical qubit (\(K=2\)) and \(t=1\), the bound becomes \(3n+1 \leq 2^{n-1}\). The bound can be saturated only at certain \(n\).

For qubit codes with \(K=2^k\), one can work out an asymptotic Hamming bound in the large-\(n,k,t\) limit, \begin{align} \frac{k}{n}\leq 1-\frac{t}{n}\log_{2}3-h(t/n), \end{align} where \(h\) is the binary entropy function.

Protection

Perfect codes have been classified. For qubits (\(q=2\)), the only codes are the stabilizer code family \([[(4^r-1)/3, (4^r-1)/3 - 2r, 3]]\) for \(r \geq 2\) [1][2]. For qudits, the family is parameterized by \(K=q^{n-2r}\) for \(n=\frac{q^{2r}-1}{q^{2}-1}\) and \(r \geq 2\); all codes correct a single error (\(t=1\)) [3][4]. The trivial code (\(k=n\)) is also perfect.

Rate

\(k/n\to 1\) asymptotically with \(n\).

Parent

Cousins

Zoo code information

Internal code ID: quantum_perfect

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: quantum_perfect

Cite as:
“Perfect quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_perfect
BibTeX:
@incollection{eczoo_quantum_perfect, title={Perfect quantum code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_perfect} }
Permanent link:
https://errorcorrectionzoo.org/c/quantum_perfect

References

[1]
Daniel Gottesman, “Pasting Quantum Codes”. quant-ph/9607027
[2]
A. R. Calderbank et al., “Quantum Error Correction via Codes over GF(4)”. quant-ph/9608006
[3]
Zhuo Li and Lijuan Xing, “No More Perfect Codes: Classification of Perfect Quantum Codes”. 0907.0049
[4]
J. Bierbrauer and Y. Edel, “Quantum twisted codes”, Journal of Combinatorial Designs 8, 174 (2000). DOI

Cite as:

“Perfect quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_perfect

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/properties/quantum_perfect.yml.