  Active quantum error correction has been identified as a crucial ingredient
of future quantum computers, motivating the recent experimental efforts to
encode logical quantum bits using small topological codes. In addition to the
demonstration of the beneficial role of the encoding, a break-even point in the
progress towards large-scale quantum computers will be the implementation of a
universal set of gates. This mid-term challenge will soon be faced by various
quantum technologies, which urges the need of realistic assessments of their
prospects. In this work, we pursue this goal by assessing the capability of
current trapped-ion architectures in facing one of the most demanding parts of
this quest: the implementation of an entangling CNOT gate between encoded
logical qubits. We present a detailed comparative study of two alternative
strategies for trapped-ion topological color codes, either a transversal or a
lattice-surgery approach, characterized by a detailed microscopic modeling of
both current technological capabilities and experimental sources of noise
afflicting the different operations. Our careful fault-tolerant design,
together with a low-resource optimization, allows us to determine via
exhaustive numerical simulations the experimental regimes where each of the
approaches becomes favorable. We hope that our study thereby contributes to
guiding the future development of trapped-ion quantum computers.
