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Optimizing the controls of quantum systems plays a crucial role in advancing quantum technologies. The time-varying noises in quantum systems and the widespread use of inhomogeneous quantum ensembles raise the need for high-quality quantum controls under uncertainties. In this paper, we consider a stochastic discrete optimization formulation of a discretized binary optimal quantum control problem involving Hamiltonians with predictable uncertainties. We propose a sample-based reformulation that optimizes both risk-neutral and risk-averse measurements of control policies, and solve these with two gradient-based algorithms using sum-up-rounding approaches. Furthermore, we discuss the differentiability of the objective function and prove upper bounds of the gaps between the optimal solutions to binary control problems and their continuous relaxations. We conduct numerical simulations on various sized problem instances based on two applications of quantum pulse optimization; we evaluate different strategies to mitigate the impact of uncertainties in quantum systems. We demonstrate that the controls of our stochastic optimization model achieve significantly higher quality and robustness compared with the controls of a deterministic model. History: Accepted by Giacomo Nannicini, Area Editor for Quantum Computing and Operations Research. Accepted for Special Issue. Funding: This work was supported by the US Department of Energy, Advanced Scientific Computing Research [Grants DE-AC02-06CH11357, DE-SC0018018]; Defense Sciences Office, DARPA [Grant IAA-8839-annex-130]; the US National Science Foundation, Division of Civil, Mechanical and Manufacturing Innovation [Grant 2041745]; and the US National Aeronautics and Space Administration (NASA) Ames Research Center [Grant 80ARC020D0010]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0560 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0560 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .