  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.
