The quasi two-dimensional electron gas (q-2DEG) at oxide interfaces provides a platform for investigating quantum phenomena in strongly correlated electronic systems. Here, we study the transport properties at the high-mobility (La$_{0.3}$Sr$_{0.7}$)(Al$_{0.65}$Ta$_{0.35}$)O$_{0.3}$/SrTiO$_{0.3}$ (LSAT/STO) interface. Before oxygen annealing, the as-grown interface exhibits a high electron density and electron occupancy of two subbands: higher-mobility electrons ($mu_1approx{10^4}$ cm$^2$V$^{-1}$s$^{-1}$ at 2 K) occupy the lower-energy $3d_{xy}$ subband, while lower-mobility electrons ($mu_1approx{10^3}$ cm$^{2}$V$^{-1}$s$^{-1}$ at 2 K) propagate in the higher-energy $3d_{xz/yz}$-dominated subband. After removing oxygen vacancies by annealing in oxygen, only a single type of 3dxy electrons remain at the annealed interface, showing tunable Shubnikov-de Haas (SdH) oscillations below 9 T at 2 K and an effective mass of $0.7m_e$. By contrast, no oscillation is observed at the as-grown interface even when electron mobility is increased to $50,000$ cm$^{2}$V$^{-1}$s$^{-1}$ by gating voltage. Our results reveal the important roles of both carrier mobility and subband occupancy in tuning the quantum transport at oxide interfaces.
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