  The impressive progress in fabricating and controlling superconducting
devices for quantum information processing has reached a level where reliable
theoretical predictions need to account for quantum correlations that are not
captured by the conventional modeling of contemporary quantum computers. This
applies particularly to the qubit initialization as the process which crucially
limits typical operation times. Here we employ numerically exact methods to
study realistic implementations of a transmon qubit embedded in electromagnetic
environments focusing on the most important system-reservoir correlation
effects such as the Lamb shift and entanglement. For the qubit initialization
we find a fundamental trade-off between speed and accuracy which sets intrinsic
constraints in the optimization of future reset protocols. Instead, the
fidelities of quantum logic gates can be sufficiently accurately predicted by
standard treatments. Our results can be used to accurately predict the
performance of specific set-ups and also to guide future experiments in probing
low-temperature properties of qubit reservoirs.
