Quantum computing is a relatively new computing paradigm that seeks to use quantum resources, like superposition and entanglement, to process information in a significantly new way. These resources allow quantum computers to solve certain problems more efficiently than their classical counterparts, with promising applications in cryptography, optimization, and materials science. This thesis investigates the development and implementation of quantum algorithms to solve certain estimation problems in quantum information science and computing. Using variational quantum algorithms and near-term hardware, we investigate three interconnected domains of research: distinguishability estimation, symmetry testing, and nuclear dynamics. In the first study, we explore the estimation of distinguishability measures, such as trace distance and fidelity, which are crucial for evaluating quantum information processing protocols. We provide novel interpretations of these different measures and study the computational complexity of the algorithms to estimate these measures. Next, we put forth several symmetry testing algorithms that estimate what we call 'maximum symmetric fidelities.' We study several different symmetry examples, including cyclicity, permutation, and others. A major contribution of this study is the connection of the symmetry testing algorithms to the computational complexity hierarchy. We provide proofs that symmetry testing algorithms are complete for different complexity classes. In the last study, we explore different qubit encoding techniques for translating nuclear physics problems to quantum computers. We analyze the various trade-offs and show that one encoding outperforms that others in all the relevant metrics. For all the studies above, we simulate the algorithms in the noiseless and noisy scenarios and show robust convergence for the examples considered.