A Novel q-Derivative Framework with Applications to q-Deformed Thermodynamics and Leakage Suppression in Superconducting Qubits

We propose a new q-derivative operator built directly from Jackson's q-number formulation, designed to preserve the structural properties of standard differential calculus while incorporating deformation effects. By analyzing q-deformed Heisenberg algebras, we demonstrate that this formulation maintains the consistency of thermodynamic quantities-such as internal energy, particle number, and specific heat-in dilute gas limits without requiring ad-hoc chain-rule modifications. Furthermore, we explore the physical implications of algebraic deformation using the Biedenharn-Macfarlane realization, showing how q-deformation induces intrinsic anharmonicity in quantum oscillator spectra and affects multi-level quantum systems (d 3). Applying this algebraic scheme to superconducting transmon qubits, we derive analytical pulse-shaping corrections that generalize the Derivative Removal by Adiabatic Gate (DRAG) technique, offering a robust method to suppress computational leakage in ultra-fast quantum logic operations.