Assessing the progress of trapped-ion processors towards fault-tolerant quantum computation

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.