Thresholds for correcting errors, erasures, and faulty syndrome   measurements in degenerate quantum codes
  We suggest a technique for constructing lower (existence) bounds for the fault-tolerant threshold to scalable quantum computation applicable to degenerate quantum codes with sublinear distance scaling. We give explicit analytic expressions combining probabilities of erasures, depolarizing errors, and phenomenological syndrome measurement errors for quantum LDPC codes with logarithmic or larger distances. These threshold estimates are parametrically better than the existing analytical bound based on percolation. 
Ilya Dumer and Alexey A. Kovalev and Leonid P. Pryadko
quant-ph