  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 have reached 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 junction or 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. In this work we extend the measurement theory of a
conventional superconducting qubit to that of a qudit, focusing on modeling the
dispersive quadrature measurement in an open quantum system. Under the Markov
assumption, the qudit Lindblad and stochastic master equations are formulated
and analyzed; in addition, both the ensemble-averaged and the quantum-jump
approach of decoherence analysis are detailed with analytical and numerical
comparisons. We verify our stochastic model with a series of experimental
results on a transmon-type qutrit, verifying the validity of our
high-dimensional formalism.
