Quantum algorithms hold the promise of fundamentally changing the way we approach problems in physics, chemistry, computer science, and engineering. This thesis explores new provably efficient quantum algorithms for three fundamental directions in quantum computing: simulation, optimization, and estimation. Together, our results in each of these directions contribute to the broader effort of making quantum computation a practical tool for science and engineering. First, in the area of quantum simulation, we propose Wave Matrix Lindbladization, a set of algorithms that extend the philosophy of density matrix exponentiation to Lindbladian dynamics by working with program-state encodings of Lindblad operators. Specifically, when these operators are local, we show that these algorithms are provably efficient by analyzing their sample and gate complexities. Second, for optimization, we study quantum Boltzmann machines as a variational ansatz for ground-state energy estimation, proving efficient gradient estimation and convergence to approximate stationary points. We also propose hybrid quantum-classical algorithms for semidefinite programming, establish rigorous convergence guarantees, and validate their robustness to noise through numerical simulations for applications such as MaxCut. Finally, for estimation, we develop the tapered quantum phase estimation algorithm, a coherent phase estimation method that leverages taper/window functions from signal processing to reach optimal query complexity without relying on expensive coherent median techniques, and we provide error guarantees in both asymptotic and non-asymptotic regimes along with efficient taper state preparation.