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 \tag*{(1)}\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), \tag*{(2)}\end{align} where \(h\) is the binary entropy function.
A quantum Hamming-like bound exists for degenerate qubit stabilizer codes [1].
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- Five-qubit perfect code — The five-qubit code is the smallest perfect code.
Cousins
- Perfect code — A classical (quantum) perfect code saturates the classical (quantum) Hamming bound.
- \([[2^r, 2^r-r-2, 3]]\) quantum Hamming code — Quantum Hamming codes saturate the asymptotic quantum Hamming bound.
- \([[15, 7, 3]]\) Hamming-based CSS code — \([[15, 7, 3]]\) Hamming-based CSS code is perfect as a CSS code, i.e., the number of its \(Z\)-type syndromes matches the number of \(X\)-type Pauli errors up to weight one [7].
References
- [1]
- A. Nemec and T. Tansuwannont, “A Hamming-Like Bound for Degenerate Stabilizer Codes”, (2023) arXiv:2306.00048
- [2]
- D. Gottesman, “Pasting Quantum Codes”, (1996) arXiv:quant-ph/9607027
- [3]
- A. R. Calderbank et al., “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
- [4]
- Z. Li and L. Xing, “No More Perfect Codes: Classification of Perfect Quantum Codes”, (2009) arXiv:0907.0049
- [5]
- J. Bierbrauer and Y. Edel, “Quantum twisted codes”, Journal of Combinatorial Designs 8, 174 (2000) DOI
- [6]
- E. Dallas, F. Andreadakis, and D. Lidar, “No \(((n,K,d< 127))\) Code Can Violate the Quantum Hamming Bound”, IEEE BITS the Information Theory Magazine 1 (2023) arXiv:2208.11800 DOI
- [7]
- R. Chao and B. W. Reichardt, “Fault-tolerant quantum computation with few qubits”, npj Quantum Information 4, (2018) arXiv:1705.05365 DOI
Page edit log
- Victor V. Albert (2022-08-26) — most recent
- Mazin Karjikar (2022-06-03)
- Victor V. Albert (2021-12-03)
Cite as:
“Perfect quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_perfect