Landsman, Kevin AntonyA large-scale quantum computer will have the ability to solve many computational problems beyond the capabilities of today's most powerful computers. Significant efforts to build such a computer are underway, many of which are small prototypes that are believed to be extensible to larger systems. Such systems, like the one in this thesis built off of 171Yb+ ions, are enticing scientific endeavors for their potential to inform the production of large-scale systems, as well as the interesting experiments they can perform. In this work, experimental research is presented on both topics: scalability as well as compelling computations. The first half of this thesis discusses building and optimizing a quantum computer to have high-fidelity qubit operations. An experimental architecture that allows for individual addressing and individual detection of qubits is presented alongside a discussion of errors and error-reduction. We derive the coherent manipulation of qubits using Raman lasers for rotational gates and the criteria necessary for multi-qubit entangling gates. Methods for efficiently fulfilling these criteria are presented with experimental data. Lastly, we consider coherence-related properties of the system necessary to perform these operations and how they can be experimentally improved. The second half of the thesis features three experimental applications of the quantum computer: quantifying quantum scrambling, applying a quantum error correction code, and measuring Renyi entropy. Quantum scrambling is the coherent delocalization of information through a quantum system and is notably difficult to quantify experimentally. We present an efficient scheme to measure it using quantum teleportation. Quantum error correction is a set of techniques for mitigating the effect of imperfect operations performed on a quantum computer, and we demonstrate some of these techniques in order to fault-tolerantly prepare a logical qubit. Lastly, \renyi entropy is an information theoretic quantity that can be used to directly quantify the amount of entanglement in a system. We present a method for measuring it efficiently using a quantum gate known as a Fredkin gate.enDissertationAtomic physicsQuantum physicsatomic physicsentanglememtion trapQuantum computingscrambling