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Quantum information science has made significant progress in recent decades. The performance of quantum hardware still remains a major bottleneck that prevents us from benefiting from the full quantum power. As we gain better control over the quantum hardware, it is crucial to use appropriate mathematical models, systematically calibrate higher order perturbative effectives, develop robust quantum control and reservoir engineering schemes, and improve numerical methods to reliably simulate complicated quantum systems. A major goal of the workshop will be to identify the underlying challenges and develop novel mathematical tools to solve these open questions.