Modeling decoherence processes in high-dimensional superconducting circuits using master equations and quantum trajectory approach*

Recent advances in Josephson-junction-based superconducting quantum computing have focused on scaling the number of logical qubits via error mitigation and corrections. Another parallel approach is to increase the size of the Hilbert space in a single device, i.e., with high-dimensional qubits, also known as qudits. It has been shown that one can drive higher-order transitions in a transmon or design new multimode superconducting circuits, called multimons, to expand the computational basis. To systematically analyze the decoherence of an artificial atom with more than two levels, we adopt an effective Lindblad master equation and stochastic master equation (SME) to model various dephasing mechanisms in a qutrit, a three-level qudit, when dispersively coupled to a readout resonator. The effectiveness of dynamical decoupling pulses on a qutrit is shown by solving the Lindblad master equation. In addition, the quantum trajectories induced by the quantum nondemolition measurement are simulated using the qutrit SME at various readout frequencies. Sample paths of the heterodyne measurements are also computed along with the quantum trajectories to find an optimal measurement time and readout frequency for multiple-state classification and to compare with real experiments conducted on a 3D transmon qutrit. We also propose that the SME can be heuristically generalized for a multimon based on the cross-Kerr matrix measured.