On the minimum distance in one class of quantum LDPC codes

We consider a family of quantum LDPC codes with weight-6 stabilizer generators and two logical qubits, where some logical operators have a fractal structure. These codes can be considered as local quantum codes on the $L \times L \times L$ cubic lattice with periodic boundary conditions. We prove that the minimum distance of codes from this family is bounded below by $\Omega (L^\alpha)$, where $\alpha = \log_2 (2(\sqrt{5} - 1)) \approx 1.306$.