Systematic study of High $E_J/E_C$ transmon qudits up to $d = 12$

Qudits provide a resource-efficient alternative to qubits for quantum information processing. The multilevel nature of the transmon, with its individually resolvable transition frequencies, makes it an attractive platform for superconducting circuit-based qudits. In this work, we systematically analyze the trade-offs associated with encoding high-dimensional quantum information in fixed-frequency transmons. Designing high $E_J/E_C$ ratios of up to 325, we observe up to 12 levels ($d=12$) on a single transmon. Despite the decreased anharmonicity, we demonstrate process infidelities $e_f < 3 \times 10^{-3}$ for qubit-like operations in each adjacent-level qubit subspace in the lowest 10 levels. Furthermore, we achieve a 10-state readout assignment fidelity of 93.8% with the assistance of deep neural network classification of a multi-tone dispersive measurement. We find that the Hahn echo time $T_{2E}$ for the higher levels is close to the limit of $T_1$ decay, primarily limited by bosonic enhancement. We verify the recently introduced Josephson harmonics model, finding that it yields better predictions for the transition frequencies and charge dispersion. Finally, we show strong $ZZ$-like coupling between the higher energy levels in a two-transmon system. Our high-fidelity control and readout methods, in combination with our comprehensive characterization of the transmon model, suggest that the high-$E_J/E_C$ transmon is a powerful tool for exploring excited states in circuit quantum electrodynamics.