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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 thebreakeven point with the novel error mitigation and correction methods.Parallel to these advances is the effort to expand the Hilbert space within asingle junction or device by employing high-dimensional qubits, otherwise knownas qudits. Research has demonstrated the possibility of driving higher-ordertransitions in a transmon or designing innovative multimode superconductingcircuits, termed multimons. These advances can significantly expand thecomputational basis while simplifying the interconnects in a large-scalequantum processor. In this work we extend the measurement theory of aconventional superconducting qubit to that of a qudit, focusing on modeling thedispersive quadrature measurement in an open quantum system. Under the Markovassumption, the qudit Lindblad and stochastic master equations are formulatedand analyzed; in addition, both the ensemble-averaged and the quantum-jumpapproach of decoherence analysis are detailed with analytical and numericalcomparisons. We verify our stochastic model with a series of experimentalresults on a transmon-type qutrit, verifying the validity of ourhigh-dimensional formalism.

Stochastic modeling of superconducting qudits in the dispersive regime