  The field of superconducting quantum computing, based on Josephson junctions,
has recently seen remarkable strides in scaling the number of logical qubits.
In particular, the fidelities of one- and two-qubit gates are close to the
breakeven point with the novel error mitigation and correction methods.
Parallel to these advances is the effort to expand the Hilbert space within a
single device by employing high-dimensional qubits, otherwise known as qudits.
Research has demonstrated the possibility of driving higher-order transitions
in a transmon or designing innovative multimode superconducting circuits,
termed multimons. These advances can significantly expand the computational
basis while simplifying the interconnects in a large-scale quantum processor.
This thesis provides a detailed introduction to the superconducting qudit and
demonstrates a comprehensive analysis of decoherence in an artificial atom with
more than two levels using Lindblad master equations and stochastic master
equations (SMEs). After extending the theory of the design, control, and
readout of a conventional superconducting qubit to that of a qudit, the thesis
focuses on modeling the dispersive measurement of a transmon qutrit in an open
quantum system using quadrature detections. Under the Markov assumption, master
equations with different levels of abstraction are proposed and solved; in
addition, both the ensemble-averaged and the quantum-jump approach of
decoherence analysis are presented and compared analytically and numerically.
The thesis ends with a series of experimental results on a transmon-type
qutrit, verifying the validity of the stochastic model.
