Quantum LDPC Codes With Almost Linear Minimum Distance
We give a construction of quantum LDPC codes of dimension <inline-formula> <tex-math notation="LaTeX">\Theta (\log N) </tex-math></inline-formula> and distance <inline-formula> <tex-math notation="LaTeX">\Theta (N/\log N) </tex-math></...
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| Veröffentlicht in: | IEEE transactions on information theory Jg. 68; H. 1; S. 213 - 229 |
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| Sprache: | Englisch |
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IEEE
01.01.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-9448, 1557-9654 |
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| Abstract | We give a construction of quantum LDPC codes of dimension <inline-formula> <tex-math notation="LaTeX">\Theta (\log N) </tex-math></inline-formula> and distance <inline-formula> <tex-math notation="LaTeX">\Theta (N/\log N) </tex-math></inline-formula> as the code length <inline-formula> <tex-math notation="LaTeX">N\to \infty </tex-math></inline-formula>. Using a product of chain complexes this construction also provides a family of quantum LDPC codes of distance <inline-formula> <tex-math notation="LaTeX">\Omega (N^{1-\alpha /2}/\log N) </tex-math></inline-formula> and dimension <inline-formula> <tex-math notation="LaTeX">\Omega (N^\alpha \log N) </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">0 \le \alpha < 1 </tex-math></inline-formula>. We also introduce and study a new operation called lifted product, which naturally generalizes the product operations for quantum codes and chain complexes. Moreover, as a simple byproduct of our results on quantum codes, we obtain a new result on classical codes. We show that for any fixed <inline-formula> <tex-math notation="LaTeX">R < 1 </tex-math></inline-formula> there exists an asymptotically good family of classical quasi-cyclic LDPC codes of rate at least <inline-formula> <tex-math notation="LaTeX">R </tex-math></inline-formula> with, in some sense, optimal circulant size <inline-formula> <tex-math notation="LaTeX">\Omega (N/\log N) </tex-math></inline-formula> as the code length <inline-formula> <tex-math notation="LaTeX">N\to \infty </tex-math></inline-formula>. |
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| AbstractList | We give a construction of quantum LDPC codes of dimension <inline-formula> <tex-math notation="LaTeX">\Theta (\log N) </tex-math></inline-formula> and distance <inline-formula> <tex-math notation="LaTeX">\Theta (N/\log N) </tex-math></inline-formula> as the code length <inline-formula> <tex-math notation="LaTeX">N\to \infty </tex-math></inline-formula>. Using a product of chain complexes this construction also provides a family of quantum LDPC codes of distance <inline-formula> <tex-math notation="LaTeX">\Omega (N^{1-\alpha /2}/\log N) </tex-math></inline-formula> and dimension <inline-formula> <tex-math notation="LaTeX">\Omega (N^\alpha \log N) </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">0 \le \alpha < 1 </tex-math></inline-formula>. We also introduce and study a new operation called lifted product, which naturally generalizes the product operations for quantum codes and chain complexes. Moreover, as a simple byproduct of our results on quantum codes, we obtain a new result on classical codes. We show that for any fixed <inline-formula> <tex-math notation="LaTeX">R < 1 </tex-math></inline-formula> there exists an asymptotically good family of classical quasi-cyclic LDPC codes of rate at least <inline-formula> <tex-math notation="LaTeX">R </tex-math></inline-formula> with, in some sense, optimal circulant size <inline-formula> <tex-math notation="LaTeX">\Omega (N/\log N) </tex-math></inline-formula> as the code length <inline-formula> <tex-math notation="LaTeX">N\to \infty </tex-math></inline-formula>. We give a construction of quantum LDPC codes of dimension [Formula Omitted] and distance [Formula Omitted] as the code length [Formula Omitted]. Using a product of chain complexes this construction also provides a family of quantum LDPC codes of distance [Formula Omitted] and dimension [Formula Omitted], where [Formula Omitted]. We also introduce and study a new operation called lifted product, which naturally generalizes the product operations for quantum codes and chain complexes. Moreover, as a simple byproduct of our results on quantum codes, we obtain a new result on classical codes. We show that for any fixed [Formula Omitted] there exists an asymptotically good family of classical quasi-cyclic LDPC codes of rate at least [Formula Omitted] with, in some sense, optimal circulant size [Formula Omitted] as the code length [Formula Omitted]. |
| Author | Panteleev, Pavel Kalachev, Gleb |
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| Snippet | We give a construction of quantum LDPC codes of dimension <inline-formula> <tex-math notation="LaTeX">\Theta (\log N) </tex-math></inline-formula> and distance... We give a construction of quantum LDPC codes of dimension [Formula Omitted] and distance [Formula Omitted] as the code length [Formula Omitted]. Using a... |
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| SubjectTerms | Binary system chain complex Chains Codecs Codes CSS code Graph theory hypergraph product code Low density parity check codes Parity check codes Product codes quantum LDPC Quantum mechanics quasi-cyclic (QC) LDPC Sparse matrices |
| Title | Quantum LDPC Codes With Almost Linear Minimum Distance |
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