  Quantum low-density parity-check (QLDPC) codes are among the most promising
candidates for future quantum error correction schemes. However, a limited
number of short to moderate-length QLDPC codes have been designed and their
decoding performance is sub-optimal with a quaternary belief propagation (BP)
decoder due to unavoidable short cycles in their Tanner graphs. In this paper,
we propose a novel joint code and decoder design for QLDPC codes. The
constructed codes have a minimum distance of about the square root of the block
length. In addition, it is, to the best of our knowledge, the first QLDPC code
family where BP decoding is not impaired by short cycles of length 4. This is
achieved by using an ensemble BP decoder mitigating the influence of assembled
short cycles. We outline two code construction methods based on classical
quasi-cyclic codes and finite geometry codes. Numerical results demonstrate
outstanding decoding performance over depolarizing channels.
