  While the accuracy of qubit operations has been greatly improved in the last
decade, further development is demanded to achieve the ultimate goal: a
fault-tolerant quantum computer that can solve real-world problems more
efficiently than classical computers. With growing fidelities even subtle
effects of environmental noise such as qubit-reservoir correlations and
non-Markovian dynamics turn into the focus for both circuit design and control.
To guide progress, we disclose, in a numerically rigorous manner, a
comprehensive picture of the single-qubit dynamics in presence of a broad class
of noise sources and for entire sequences of gate operations. Thermal
reservoirs ranging from Ohmic to deep $1/f^{\varepsilon}$-like sub-Ohmic
behavior are considered to imitate realistic scenarios for superconducting
qubits. Apart from dynamical features, two figures of merit are analyzed,
namely, fidelities of the qubit performance over entire sequences and coherence
times in presence of quantum control schemes such as the Hahn echo and
dynamical decoupling. The relevance of retarded feedback and long-range
qubit-reservoir correlations is demonstrated on a quantitative level, thus,
providing a deeper understanding of the limitations of performances for current
devices and guiding the design of future ones.
