The standard toolbox of modeling and characterizing quantum systems comes with a standard set of assumptions as well. The two-level approximation replaces a many-level system with a qubit, and the Markovian approximation assumes an environment with a short memory. In this thesis, these assumptions are relaxed, and the dynamics of a single qutrit are reconstructed from a complete set of measurement data, using maximum-likelihood estimation (MLE) to self-consistently infer a set of state-preparation and measurement parameters (SPAM), along with a time-dependent process map. The process map can then be used to quantify the non-Markovianity of the qutrit evolution. The SPAM parameters and process maps produced by the MLE framework are compared to ground-truth simulations, with good agreement found in all cases studied. A Markovian example, the amplitude and phase damping channel, and a non-Markovian example, two transmons with a static coupling, are investigated. With its ability to directly capture higher level effects such as leakage errors, and also to detect non-completely positive evolution due to entanglement with the environment, this framework improves upon existing characterization algorithms with the purpose of encouraging future experimental work with qutrits.
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