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Quantum computation appears to offer significant advantages over classicalcomputation and this has generated a tremendous interest in the field. In thisthesis we consider the application of quantum computers to scientific computingand combinatorial optimization. We study five problems. The first three dealwith quantum algorithms for computational problems in science and engineering,including quantum simulation of physical systems. In particular, we studyquantum algorithms for numerical computation, for the approximation of groundand excited state energies of the Schr\"odinger equation, and for Hamiltoniansimulation with applications to physics and chemistry. The remaining two dealwith quantum algorithms for approximate optimization. We study the performanceof the quantum approximate optimization algorithm (QAOA), and show ageneralization of QAOA, the $\textit{quantum}$ $\textit{alternating}$$\textit{operator}$ $\textit{ansatz}$, particularly suitable for constrainedoptimization problems and low-resource implementations on near-term quantumdevices.

Quantum Algorithms for Scientific Computing and Approximate Optimization