Quantum LDPC Codes With Almost Linear Minimum Distance

• Published in: IEEE transactions on information theory Vol. 68; no. 1; pp. 213 - 229
• Main Authors: Panteleev, Pavel, Kalachev, Gleb
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.01.2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: 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
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2021.3119384

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>.