Book excerpt: Quantum computing offers computational tools and speed-ups beyond the reach of classical computers in the long run, yet demonstrating the computational advantage of small-scale near-term quantum devices remains a challenge today. In this thesis, I introduce two applications of near-term quantum computers from the fields of spin chemistry and combinatorial optimization. In the first part of my thesis, I discuss the quantum beats phenomena that occur in spin chemistry when geminate radical pairs go through singlet-to-triplet conversions under hyperfine coupling interactions and an external magnetic field. I demonstrate that both the coherent time evolution and the thermal relaxation of these radical pair systems can be efficiently simulated on a quantum computer, both hardware and simulator. I introduce the first Hamiltonian simulation method on a quantum computer that can simulate the quantum beats phenomenon for radical pair systems with nontrivial hyperfine interactions under an arbitrary magnetic field, by utilizing Hamiltonian partitioning strategies. The second part of my thesis focuses on the open-pit mining problem, where the optimal mine digging pattern that maximizes the profit is searched for, while obeying slope constraints. This optimization is traditionally solved as a maximum closure problem. Our algorithm formulates this task as a Hamiltonian ground state search in order to map it to a quantum computer, and utilizes VQE as a subroutine to find the optimal solution. I show that the Hamiltonian can be decomposed into smaller, classically correlated partitions that can each be iteratively optimized. This will allow us to make use of near-term quantum hardware for larger scale optimization problems.