  Control of multi-level quantum systems is sensitive to implementation errors
in the control field and uncertainties associated with system Hamiltonian
parameters. A small variation in the control field spectrum or the system
Hamiltonian can cause an otherwise optimal field to deviate from controlling
desired quantum state transitions and reaching a particular objective. An
accurate analysis of robustness is thus essential in understanding and
achieving model-based quantum control, such as in control of chemical reactions
based on ab initio or experimental estimates of the molecular Hamiltonian. In
this paper, theoretical foundations for quantum control robustness analysis are
presented from both a distributional perspective - in terms of moments of the
transition amplitude, interferences, and transition probability - and a
worst-case perspective. Based on this theory, analytical expressions and a
computationally efficient method for determining the robustness of coherently
controlled quantum dynamics are derived. The robustness analysis reveals that
there generally exists a set of control pathways that are more resistant to
destructive interferences in the presence of control field and system parameter
uncertainty. These robust pathways interfere and combine to yield a relatively
accurate transition amplitude and high transition probability when uncertainty
is present.
