Quantum information science has increased research toward using superconducting circuits as qubits in quantum information processors. Superconducting qubits can strongly and controllably interact with microwave photons leading to circuit quantum electrodynamics (QED), the topic of this thesis. Circuit QED is essential in understanding the interaction of quantum systems with microwave photons. Furthermore, circuit QED is benecial for error mitigation, entanglement, and single-photon generation used in quantum computation. This thesis overviews the fundamental concepts of the quantization of the LC oscillator and transmission line theory. This thesis also explores the coupling of a case study superconducting circuit to the environment in an open quantum system. Understanding this coupling is necessary for coherent control, measurement and readout. The next chapter overviews the fundamentals of multiconductor transmission line theory that has applications in more extensive qubit networks and modeling three-wave and four-wave mixing properties through frequency matching. This thesis then outlines the formalisms for a ux-pumped Superconducting QUantum Interference Device (SQUID) for the three-wave and four-wave degenerate and non-degenerate cases. The ux-pumped SQUID allows for signal amplication with minimal added noise. Finally, this thesis proposes using a multiconductor transmission line network to model the three-wave ux-pumped SQUID.
