  Fault-tolerant quantum error correction provides a strategy to protect
information processed by a quantum computer against noise which would otherwise
corrupt the data. A fault-tolerant universal quantum computer must implement a
universal gate set on the logical level in order to perform arbitrary
calculations to in principle unlimited precision. We characterize the recent
demonstration of a fault-tolerant universal gate set in a trapped-ion quantum
computer [Postler et al. Nature 605.7911 (2022)] and identify aspects to
improve the design of experimental setups to reach an advantage of logical over
physical qubit operation. We show that various criteria to assess the
break-even point for fault-tolerant quantum operations are within reach for the
ion trap quantum computing architecture under consideration. We analyze the
influence of crosstalk in entangling gates for logical state preparation
circuits. These circuits can be designed to respect fault tolerance for
specific microscopic noise models. We find that an experimentally-informed
depolarizing noise model captures the essential noise dynamics of the
fault-tolerant experiment, and crosstalk is negligible in the currently
accessible regime of physical error rates. For deterministic Pauli state
preparation, we provide a fault-tolerant unitary logical qubit initialization
circuit, which can be realized without in-sequence measurement and feed-forward
of classical information. We show that non-deterministic state preparation
schemes for logical Pauli and magic states perform with higher logical fidelity
over their deterministic counterparts for the current and anticipated future
regime of physical error rates. Our results offer guidance on improvements of
physical qubit operations and validate the experimentally-informed noise model
as a tool to predict logical failure rates in quantum computing architectures
based on trapped ions.
