Quantum LDPC Codes With Almost Linear Minimum Distance

We give a construction of quantum LDPC codes dimension $\Theta(\log N)$ and distance $\Theta(N/\log as the code length $N\to\infty$. Using product chain complexes this also provides family $\Omega(N^{1-\alpha/2}/\log $\Omega(N^\alpha \log N)$, where $0 \le \alpha < 1$. introduce study new operation called lifted product, which naturally generalizes operations for complexes. Moreover, simple byproduct our results on codes, we obtain result classical codes. show that any fixed $R 1$ there exists an asymptotically good quasi-cyclic rate at least $R$ with, in some sense, optimal circulant size $\Omega(N/\log