  A quantitative assessment of the progress of small prototype quantum
processors towards fault-tolerant quantum computation is a problem of current
interest in experimental and theoretical quantum information science. We
introduce a necessary and fair criterion for quantum error correction (QEC),
which must be achieved in the development of these quantum processors before
their sizes are sufficiently big to consider the well-known QEC threshold. We
apply this criterion to benchmark the ongoing effort in implementing QEC with
topological color codes using trapped-ion quantum processors and, more
importantly, to guide the future hardware developments that shall be required
in order to demonstrate beneficial QEC with small topological quantum codes. In
doing so, we present a thorough description of a realistic trapped-ion toolbox
for QEC, and a physically-motivated error model that goes beyond standard
simplifications in the QEC literature. Our large-scale numerical analysis shows
that two-species trapped-ion crystals in high-optical aperture segmented traps,
with the improvements hereby described, are a very promising candidate for
fault-tolerant quantum computation.
