  An arbitrarily reliable quantum computer can be efficiently constructed from
noisy components using a recursive simulation procedure, provided that those
components fail with probability less than the fault-tolerance threshold.
Recent estimates of the threshold are near some experimentally achieved gate
fidelities. However, the landscape of threshold estimates includes
pseudothresholds, threshold estimates based on a subset of components and a low
level of recursion. In this paper, we observe that pseudothresholds are a
generic phenomenon in fault-tolerant computation. We define pseudothresholds
and present classical and quantum fault-tolerant circuits exhibiting
pseudothresholds that differ by a factor of 4 from fault-tolerance thresholds
for typical relationships between component failure rates. We develop tools for
visualizing how reliability is influenced by recursive simulation in order to
determine the asymptotic threshold. Finally, we conjecture that refinements of
these methods may establish upper bounds on the fault-tolerance threshold for
particular codes and noise models.
