  Many proposals for quantum information processing are subject to detectable
loss errors. In this paper, we give a detailed account of recent results in
which we showed that topological quantum memories can simultaneously tolerate
both loss errors and computational errors, with a graceful tradeoff between the
threshold for each. We further discuss a number of subtleties that arise when
implementing error correction on topological memories. We particularly focus on
the role played by degeneracy in the matching algorithms, and present a
systematic study its effects on thresholds. We also discuss some of the
implications of degeneracy for estimating phase transition temperatures in the
random bond Ising model.
