Quantum Algorithms for Scientific Computing and Approximate Optimization

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