Building a scalable quantum computer is one of the greatest challenges of this century. A quantum computer should be able to execute quantum algorithms which promise great speed up in solving certain computational problems.Microwave photons in superconducting circuits offer a promising platform for quantum computing due to their long coherence times, and the ease of their generation and control. Single-qubit quantum gates are implemented with conventional microwave components. However, implementing entangling two-qubit gates for photonic qubits is considerably difficult because photons do not interact with each other naturally.In this thesis, we investigate the effective interactions between photons mediated by superconducting qubits. Especially, the nonlinear effects in the photonic states generated by these interactions are studied.Based on the analysis of the scattering properties of the system, we present a design for a nonlinear phase shifter device, which yields a photon-number-dependent phase shift in the transmitted photons. Furthermore, we show how entangling two-qubit gates can be implemented using the presented nonlinear phase shifter.
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