Description
A type of block quantum code whose parameters satisfy the quantum Hamming bound with equality.
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 [1], \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 such codes. 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.
Degenerate codes can in principle violate the quantum Hamming bound. It was shown that qubit stabilizer codes correcting up to two errors [2], qudit stabilizer codes up to distance two [3], qudit CSS codes of qudit dimension \(q\geq 5\) along with certain other codes [4], and qubit codes up to distance \(d\leq 127\) [5] do not violate the bound. A quantum Hamming-like bound exists for degenerate qubit stabilizer codes [6].
Protection
Perfect codes have been classified. For qubits (\(q=2\)), the only nontrivial codes are the stabilizer code family \([[(4^r-1)/3, (4^r-1)/3 - 2r, 3]]\) for \(r \geq 2\), obtained from Hamming codes over \(GF(4)\) via the Hermitian construction [7,8]. For qudits, the corresponding family is the \([[\frac{q^{2r}-1}{q^{2}-1},q^{n-2r},3]]_q\) family of quantum twisted codes [9,10].Rate
\(k/n\to 1\) asymptotically with \(n\).Notes
Cousins
- Perfect code— A classical (quantum) perfect code saturates the classical (quantum) Hamming bound.
- Hermitian qubit code— The only perfect qubit codes are the Hermitian qubit code family \([[(4^r-1)/3, (4^r-1)/3 - 2r, 3]]\) for \(r \geq 2\), obtained from Hamming codes over \(GF(4)\) [7,8].
- \([[2^r, 2^r-r-2, 3]]\) Gottesman code— \([[2^r, 2^r-r-2, 3]]\) Gottesman codes saturate the asymptotic quantum Hamming bound.
- Quantum data-syndrome (QDS) code— The quantum Hamming bound can be extended to QDS codes [11].
- \([[15, 7, 3]]\) quantum Hamming code— \([[15, 7, 3]]\) quantum Hamming 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 [12].
- Modular-qudit CWS code— Generalized concatenatenations of modular-qudit CWS codes yield a family of codes that have larger logical dimension than stabilizer codes and that asymptotically approach the modular-qudit Hamming bound [13].
- Modular-qudit GKP code— The modular-qudit GKP code is not a block code, but it is perfect in the sense that each correctable error maps the logical space into a distinct error space.
- Modular-qudit shift-resistant code— The modular-qudit shift-resistant code is not a block code, but it is perfect in the sense that each correctable error maps the logical space into a distinct error space [14].
- Quantum twisted code— The \([[\frac{q^{2r}-1}{q^{2}-1},q^{n-2r},3]]_q\) family of quantum twisted codes are the only perfect Galois-qudit codes [9,10].
Primary Hierarchy
References
- [1]
- A. Ekert and C. Macchiavello, “Error Correction in Quantum Communication”, (1996) arXiv:quant-ph/9602022
- [2]
- D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
- [3]
- S. A. Aly, “A Note on Quantum Hamming Bound”, (2007) arXiv:0711.4603
- [4]
- D. W. Kribs, A. Pasieka, and K. Zyczkowski, “Entropy of a quantum error correction code”, (2008) arXiv:0811.1621
- [5]
- 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 2, 33 (2022) arXiv:2208.11800 DOI
- [6]
- A. Nemec and T. Tansuwannont, “A Hamming-Like Bound for Degenerate Stabilizer Codes”, (2023) arXiv:2306.00048
- [7]
- D. Gottesman, “Pasting Quantum Codes”, (1996) arXiv:quant-ph/9607027
- [8]
- A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
- [9]
- Z. Li and L. Xing, “No More Perfect Codes: Classification of Perfect Quantum Codes”, (2009) arXiv:0907.0049
- [10]
- J. Bierbrauer and Y. Edel, “Quantum twisted codes”, Journal of Combinatorial Designs 8, 174 (2000) DOI
- [11]
- 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
- [12]
- R. Chao and B. W. Reichardt, “Fault-tolerant quantum computation with few qubits”, npj Quantum Information 4, (2018) arXiv:1705.05365 DOI
- [13]
- M. Grassl, P. Shor, G. Smith, J. Smolin, and B. Zeng, “Generalized concatenated quantum codes”, Physical Review A 79, (2009) arXiv:0901.1319 DOI
- [14]
- S. Pirandola, S. Mancini, S. L. Braunstein, and D. Vitali, “Minimal qudit code for a qubit in the phase-damping channel”, Physical Review A 77, (2008) arXiv:0705.1099 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