  Robustness of quantum operations or controls is important to build reliable
quantum devices. The robustness-infidelity measure (RIM$_p$) is introduced to
statistically quantify the robustness and fidelity of a controller as the
p-order Wasserstein distance between the fidelity distribution of the
controller under any uncertainty and an ideal fidelity distribution. The
RIM$_p$ is the p-th root of the p-th raw moment of the infidelity distribution.
Using a metrization argument, we justify why RIM$_1$ (the average infidelity)
suffices as a practical robustness measure. Based on the RIM$_p$, an
algorithmic robustness-infidelity measure (ARIM) is developed to quantify the
expected robustness and fidelity of controllers found by a control algorithm.
The utility of the RIM and ARIM is demonstrated by considering the problem of
robust control of spin-$\tfrac{1}{2}$ networks using energy landscape shaping
subject to Hamiltonian uncertainty. The robustness and fidelity of individual
control solutions as well as the expected robustness and fidelity of
controllers found by different popular quantum control algorithms are
characterized. For algorithm comparisons, stochastic and non-stochastic
optimization objectives are considered, with the goal of effective RIM
optimization in the latter. Although high fidelity and robustness are often
conflicting objectives, some high fidelity, robust controllers can usually be
found, irrespective of the choice of the quantum control algorithm. However,
for noisy optimization objectives, adaptive sequential decision making
approaches such as reinforcement learning have a cost advantage compared to
standard control algorithms and, in contrast, the infidelities obtained are
more consistent with higher RIM values for low noise levels.
