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Abstract
Continued improvements to superconducting architectures have placed them as one of the leading quantum systems. In particular, circuit quantum electrodynamics (cQED) combines the strong interactions and controllability of superconducting circuits with the high coherence times of 3D cavities. These features readily allow for the implementation a hardware-efficient quantum processor, even with only a single nonlinear device. In this thesis, we explore one such system that consists of a superconducting transmon qubit coupled to a 3D multimode flute cavity that supports 9 modes with qubit-mode cooperativities exceeding 1 billion and mode lifetimes $\sim$2 ms. First, we describe the system and demonstrate how to characterize it, determining the parameters necessary to effectively generate and manipulate quantum states. Next, we expand the cQED toolbox by extending state preparation techniques like photon blockade and quantum optimal control to multiple modes, implementing qudits and a pure multimodal interaction that can engineer the modes' Hilbert space. We use this interaction to easily generate entangled W states, which we demonstrate for up to 5 modes. Finally, to accurately characterize our complex states, we develop a generalized Wigner tomography method that functions even with imperfect parity measurements. We also introduce a new Wigner tomography sampling method whose required number of observations scales polynomially with subspace size, avoiding the traditional exponential scaling with mode number for certain states and thereby allowing for much more efficient state reconstruction. The developments presented in this work enables further study of complicated quantum states, especially multimode ones, with applications in quantum simulation, quantum optics, and quantum information.