This thesis explores the challenges and opportunities in quantum computing, focusing on mitigating current limitations and advancing quantum simulation techniques. After introducing the fundamentals of quantum simulation, superconducting circuits, and error mitigation we turn to applications of randomized quantum algorithms, emphasizing on dynamics. Key contributions include: improving initial state preparation using the Variational Quantum Eigensolver, extending the capabilities of the qDrift random product formula, and proposing cost-reduction algorithms for quantum simulations. Applications of these methods include constructing stochastic channels for computing the flavor variance of forward scattering neutrinos and observing the Kibble-Zurek mechanism when driving across a quantum phase transition. The thesis culminates with a framework based on Fourier moments, which facilitates arbitrary Hamiltonian transformations with applications in response functions and ground state energy estimation. We focus on the practical side of the algorithm and discuss how to solve the related challenges using techniques from signal processing, leading to a 10 fold reduction in sample complexity. Combining this framework with DMRG, we demonstrate accurate ground state energy estimation for systems of interacting neutrinos. While practical quantum advantage remains out of reach, this work lays the foundation for improved algorithms and error mitigation techniques, contributing to the progress needed to unlock the full potential of quantum computing.